Predicting the metabolic condition of a cell culture

ABSTRACT

A method for predicting the metabolic state of a cell culture of cells of a specific cell type includes providing a metabolic model of a cell of the specific cell type, and performing at each of a plurality of points in time during cultivation of the cell culture, receiving measured concentrations of a plurality of extracellular metabolites and a measured cell density in the culture medium; inputting the received measurements as input parameter values to a trained machine learning program logic—MLP; predicting extracellular fluxes of the extracellular metabolites at a future point in time by the MLP; performing metabolic flux analysis to calculate the intracellular fluxes at the future point in time based on the predicted extracellular fluxes and the stoichiometric equations of the metabolic model.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT International Application No.PCT/EP2019/050006 which has an International filing date of Jan. 2,2019, which claims priority to European Patent Application No.17211217.9, filed Dec. 29, 2017, the entire contents of each of whichare hereby incorporated by reference.

DESCRIPTION Field of Invention

The invention relates to the prediction of the metabolic state of cells,in particular of cells maintained in cell culture.

Background and Related Art

In recent years, the pharmaceutical industry has seen a significantincrease in efforts to make processes in development, production andquality assurance more efficient by placing a stronger focus on means ofprocess analysis, monitoring and control. This tendency affects also andto a special extent the operation of cell culture reactors, which areused to produce pharmaceutically relevant molecules, especiallymacromolecules such as proteins.

The modelling of cell metabolism and fermentation processes in thecontext of pharmaceutical research and development, however, turns outto be a major technical challenge: The metabolism of cells, inparticular of eukaryotic cells, is characterized by very complexnon-linear chemical reaction cascades which are difficult toanalytically simulate or model.

For example, the analysis of metabolic fluxes using metabolic fluxanalysis (MFA) is well known. MFA is particularly used for processes inwhich the intracellular flux distribution is time-invariant. This isapproximately given for the exponential growth phase in batchbioreactors or for cultivation in chemostats. However, in the fed-batchprocesses predominantly used today, the cells are exposed to constantlychanging environmental conditions. Therefore, the intracellular fluxesvary during the processes. Therefore, it is not sufficient to perform asingle metabolic flow analysis to map the state of the cells throughoutthe entire process. In addition, it is unfavorable with regard toprocess control if it is only descriptively determined that the cells ofa bioreactor have already entered a metabolically unfavorable state. Itwould be better to predict this and, if necessary, take countermeasuresat an early stage. The use of MFA for the analysis of metabolic fluxesis described in Ahn W S, Antoniewics M R (2012): “Towards dynamicmetabolic flux analysis in CHO cell cultures”, Biotechnology Journal 7,61-74.

However, the attempt to predict the state of the cells in a cell cultureat a future point in time is also very difficult for several reasons.Kinetic models are usually used to predict the course of time. Theseconsist of a system of differential equations which describe the changesover time of substance concentrations or of substance quantities. Theequations are usually based on mechanistic knowledge, which can bemodelled, for example, using Michaelis-Menten kinetics. However, thiskinetic knowledge is difficult to obtain. For the description of anentire intracellular metabolic network using kinetic expressions, theavailable mechanistic information is usually not sufficient and thenumber of parameters to be estimated would be enormous.

However, kinetic hybrid models that combine metabolic flux analysis withkinetic knowledge were in practice mostly not able to generate reliablepredictions of the future behaviour of a cell culture. The generation ofthese models has proven to be extremely labor-intensive. Moreover, themodels generated in this way are not very flexible and cannot be adaptedto the metabolism of other cell types without considerable manualeffort. The use of hybrid models is described for example in Covert M W,Xiao N, Chen T J, Karr J R (2008), “Integrating metabolic,transcriptional regulatory and signal transduction models in Escherichiacoli”, Bioinformatics Vol. 24 no. 18, 2044, and in Nolan R P, Lee K(2011) “Dynamic model of CHO cell metabolism”, Metabolic Engineering 13,108.

SUMMARY

In this context, there is a need for improved methods for predicting themetabolic state of cells and for correspondingly improved systems to theextent that the above-mentioned disadvantages are at least partiallyavoided.

The subject matter of the invention is stated in the independent claims.The embodiments of the invention are described in the dependent claims.The embodiments and examples of the invention described here may befreely combined with each other, provided they are not mutuallyexclusive.

In one aspect, the invention relates to a method for predicting themetabolic state of a cell culture of cells of a specific cell type. Themethod comprises providing a metabolic model of a cell of the specificcell type, wherein the metabolic model includes a plurality ofintracellular and extracellular metabolites and a plurality ofintracellular and extracellular fluxes, and wherein the metabolic modelcomprises stoichiometric equations specifying at least onestoichiometric relationship between one of the intracellular and one ofthe extracellular metabolites.

The method further comprises performing the following steps at each of anumber of points in time during the cultivation of the cell culture:

-   -   receiving a plurality of measurement values measured at that        point in time, the measurement values comprising concentrations        of a plurality of extracellular metabolites of the metabolic        model in the culture medium of the cell culture and a measured        cell density of the cells in the cell culture;    -   inputting the received measured values as input parameter values        into a trained machine learning program logic (MLP);    -   predicting extracellular fluxes of the extracellular metabolites        at a future point in time by the MLP using the received        measurement values, the future point in time being a point in        time subsequent to the point in time of receiving the        measurement values, wherein the extracellular fluxes are uptake        rates of the extracellular metabolites into a cell and/or        release rates of the extracellular metabolites from a cell into        the medium;    -   performing metabolic flux analysis to calculate the        intracellular fluxes at the future point in time using the        predicted extracellular fluxes of the extracellular metabolites        and the stoichiometric equations of the metabolic model.

This may be advantageous because in this way the knowledge, which forexample in the form of stoichiometric reaction equations is alreadyavailable in the literature for many important metabolic processes ofcells, and which can be represented in the form of metabolic models, canbe combined in a very advantageous way with the use of an MLP, whichallows a high degree of flexibility. Some preliminary experimentalstudies have shown that the use of MLP-based approaches, for example theuse of neural networks to predict future metabolic states of cellspurely based on, for example, the concentration of their excretionproducts, is unreliable. The complexity and dynamics of cellularmetabolism seem to make it impossible even for sophisticated, modern MLPmethods to provide a reliable prediction of the metabolic state of acell culture in the future purely based on current cell concentrations,especially in fed-batch bioreactors with eukaryotic cells. On the onehand, this might be due to the fact that the internal cell state is onlyvery indirectly and insufficiently characterized by the measurement ofextracellular metabolite concentrations. On the other hand, there isalso the risk of “over-fitting” to the training data in the course ofgenerating the MLP in the training phase. However, it has been shownthat MLP-based methods, as described here for training forms, mayprovide very reliable predictions regarding the concentration and fluxesof extracellular metabolites at least for a point in time not too far inthe future. By combining this knowledge with the knowledge ofstoichiometric and transport dependencies between intracellular andextracellular metabolites and between each other, embodiments of theinvention enables to accurately predict the metabolic state of the cellsdown to the level of individual intracellular fluxes.

Although a coarse characterization of the metabolic state of cells basedon the substances excreted by them (such as lactate) has been possibleup to now, the internal processes of the cell ultimately remained a“black box”. A fine-granular prediction of cell metabolism down to thelevel of individual cell-internal material fluxes was not possible withthe currently known hybrid models.

In another advantageous aspect of the invention, a highly flexiblemethod for predicting cell metabolism is provided. It has been shownthat even by culturing a few cell cultures with the cells of a specificcell type, a sufficiently large training data set may be obtained totrain an MLP which is capable of making reliable predictions regardingthe metabolic state at a future point in time for cell cultures of thatcell type. The acquisition of the training data and the subsequenttraining of the MLP may be largely automated. In contrast, the metabolicmodel used for metabolic flux analysis, which may require extensiveliterature study and manual adaptation steps, may often be reused forcells of different cell types. Thus, in contrast to the use of hybridmodels known in the state of the art, it is often possible to provide aprediction method that takes into account specific metaboliccharacteristics of a specific cell type without the need for the complexcreation of new models, simply by cultivating a few cell cultures togenerate a training data set and then training an MLP on this trainingdata set.

In another advantageous aspect, MFA is now successfully used to predictintracellular fluxes. Up to now, MFA has mainly been used for processeswhere the intracellular flux distribution is time-invariant. This isapproximately given for the exponential phase in batch processes or forcultivation in chemostats. However, in the fed-batch processes andperfusion-like processes with cell retention that are predominantly usedtoday, the cells are exposed to environmental conditions that areconstantly changing. Therefore, the intracellular fluxes also varyduring the process. Approaches that have tried to map the state of thecells throughout the entire process have failed. In contrast, anapproach was found to accurately predict the metabolic state offed-batch reactors at least for the coming hours and days by defining adynamic metabolic model that describes the intracellular fluxes as afunction of the (possibly discretized) process time.

In a further advantageous aspect, embodiments of the method according tothe invention make it possible to control the quality of the predictionor the intermediate results obtained during this prediction during thecultivation of the cell culture. The method described above includes theprediction of extracellular fluxes. These can be easily determined fromthe change in concentration of extracellular metabolites duringfermentation by repeated sampling and analysis of samples of the culturemedium. Thus, the prediction of MLP may be checked during fermentationby repeatedly comparing the predicted and measured extracellular fluxes.If significant differences occur, the calculation may be stoppedimmediately and the causes of the deviation investigated. Otherstate-of-the-art metabolic flux prediction methods use 13C measurementdata in the steady state to predict intracellular and extracellularmaterial fluxes. These methods therefore require 13C-labelled analysesand experiments under steady-state conditions to directly predict intra-and extracellular fluxes. 13C analyses are complex and during operationof a fermenter, stationary conditions may not be assumed. In addition,the predicted intracellular fluxes cannot be empirically verified, sothere is no intermediate step in such procedures that may be comparedwith real measured data.

According to the embodiments of the invention, the MLP receives inputdata in the form of extracellular concentrations of extracellularmetabolites at a current point in time to predict the correspondingextracellular fluxes at a future point in time (next sampling time)under dynamic fermentation conditions. The intercellular MFA iscalculated based on the stoichiometric metabolism model (for the futureand optionally also the current sampling time). This allows to verifythe prediction during fermentation by measured extracellular metaboliteconcentrations.

Embodiments of the invention thus provide a method for predicting themetabolic state of cell cultures which is both accurate (since itprovides detailed information about intracellular metabolic processesand it has been shown experimentally that the predictions regardingextracellular fluxes correspond very well to the actually measuredextracellular fluxes) and which is very flexible since it may be easilyadapted to cells of another cell type.

According to embodiments, at least some of the extracellular andintracellular fluxes of the metabolic model are not based on adecomposition of a known metabolic network into elementary flux modes.In the literature, simple intracellular network fluxes have already beenestimated using MLPs. First, the elementary flux modes of the networkwere generated. Elementary flux modes represent a set of permissibleflux distributions from which all other possible flux distributions maybe combined by different weightings. The MLP outputs the correspondingweightings. However, the number of these elementary flux modes is solarge for most biochemical networks that an effective and robustestimation of the weightings is not possible.

However, according to the invention, only the extracellular fluxes vianeural networks or other MLPs are estimated. The coupling to metabolicmaterial flux analysis then established the link to intracellular fluxdistribution.

According to the embodiments of the invention, the machine learningprogram logic is thus designed to selectively predict only extracellularfluxes, but not intracellular fluxes.

This may be advantageous because the prediction is thus limited tovalues that can easily be measured even during operation of abioreactor, so that the extracellular fluxes represent intermediatevalues for the prediction of intracellular fluxes, which can easily becompared with real measured data to quickly identify errors in the MLP.

In another aspect, the method may be advantageous for the followingreasons: The metabolism of the cells depends largely on the conditionsin the bioreactor. If factors such as pH value, pO2 value, pressure andtemperature are kept constant, the metabolite concentrations in thereaction medium are particularly important for the behavior of thecells, which is why these were chosen as inputs for the MLP. Theprediction of future metabolite fluxes has two major advantages over theprediction of future metabolite concentrations at the next samplingtime:

On the one hand, the MLP trained in this way is more flexible withregard to the choice of time intervals. If networks with metaboliteconcentrations are trained as outputs, the prediction that may be madewith such a trained MLP always refers to the same time interval as inthe training data set. Already when generating the training data, itshould therefore be ensured that the intervals are chosen as uniformlyas possible, as otherwise inconsistencies may arise. If, on the otherhand, flows are predicted, this results in a certain independence of thetime interval for the current prediction from the time intervals usedduring MLP training. The method is therefore more robust and flexiblewith regard to the choice of time intervals for the prediction.

On the other hand, the metabolite concentration depends on continuousand/or pulse-like nutrient dosages, which may be handled variably insome cases. However, this variability does not exist in the predictionif the trained MLP does not treat the added doses, which were carriedout in the training data set, separately but learns blindly. By usingthe extracellular metabolite fluxes, the future concentrations may beextrapolated using equation 4.4 of the appendix. There is someflexibility with regard to the time interval, provided that the fluxesare not subject to strong fluctuations in relation to the samplingintervals. According to preferred embodiments, the time interval betweenpoints in time used for the prediction of extracellular fluxes is lessthan or equal to, or at most 20% greater than, the time intervals usedfor the preparation of the training data set.

According to embodiments, the calculated extracellular fluxes arecalculated so that they are adjusted for substance additions (e.g.glucose additions). Thus, when predicting the one or more extracellularfluxes, the information that a specific extracellular metabolite will beadded in the next time interval may be used to adjust the prediction ofthe concentrations of this extracellular metabolite accordingly. It isalso possible to adjust the feeding appropriately based on theprediction, if the MLP predicts that a deficiency of this extracellularmetabolite is to be expected. It should be noted, however, that addeddoses may generally influence the mean flux, as this results in alteredconcentrations in the reactor. If the dosage profile deviatessignificantly from the training data, inaccuracies may result.

According to embodiments of the invention, the temporal and quantitativeprofile of the feeds of metabolites to a cell culture whose metabolicstate is to be predicted is chosen to be identical or similar to thefeed profile used to generate the training data with one or moretraining bioreactors, the MLP having been trained on these trainingdata.

According to embodiments of the invention, the measurements for theacquisition of the measured data as well as the prediction of theextracellular fluxes are carried out in real time, i.e. during theoperation of the bioreactor containing the cell culture. The timebetween the collection of the measurement data and the prediction of themetabolic state of the cells is typically small and in the range of afew seconds or minutes, typically less than 15 minutes, whereas the timeintervals for individual predictions are typically in the range of 1-48hours and in particular 6-24 hours.

In a further advantageous aspect, embodiments of the invention thusenable the prediction of the metabolic state of a cell culture in realtime, since the information base on which the prediction is basedconsists of the already defined metabolic model, the already existingtrained MLP and measurement data, which may be easily collected in realtime. For example, by regularly taking a sample from the culture mediumand determining the cell count and metabolite concentration in thissample, the measurement data required to perform the prediction may beobtained.

According to embodiments of the invention, the method for predicting themetabolic state of a cell culture is carried out at a future point intime in real time continuously during the operation of a bioreactorcontaining the cell culture.

According to embodiments, the method further comprises MLP generation bymachine learning.

The generation of the MLP comprises generating a training data set,wherein the generation of the training data set comprises performing thefollowing operations at each of a plurality of training points in timeduring the cultivation of at least one training cell culture of cells ofthe specific cell type:

-   -   receiving a plurality of measurement values measured at said        training point in time, said measurement values comprising        concentrations of a plurality of extracellular metabolites of        the metabolic model in the culture medium of said at least one        training cell culture and a measured cell density of the cells        in said at least one training cell culture;    -   receiving the time indication of the current training point in        time; and    -   calculating extracellular fluxes of the extracellular        metabolites as a function of the measured values received at        that point in time and the measured values received at the        respective preceding point in time, wherein the extracellular        fluxes are uptake rates of the extracellular metabolites into        the cell and/or release rates of the extracellular metabolites        into the medium;    -   training the MLP, wherein the training comprises:        -   inputting the measured values received at each of the            training points in time as input parameter values to the            MLP, and inputting the extracellular fluxes calculated for            that following point in time at each point in time following            that training point in time as output parameter values            associated with those input parameter values to the MLP; and        -   performing a learning process by the MLP in such a way that            the MLP learns to predict the respective associated output            parameter values based on the input parameter values;    -   storing the trained MLP in a volatile or non-volatile storage        medium.

The use and generation of an MLP according to embodiments of theinvention may be advantageous, since on the one hand a high accuracy ofthe prediction can be achieved when choosing suitable machine learningalgorithms, and on the other hand an adaptation to the metabolicconditions in other cell types can be carried out very easily andwithout major manual effort or literature study. The operation of sometraining cell cultures with continuous collection and recording of theabove-mentioned measurement data and the extracellular fluxes calculatedfrom them is sufficient to provide a training data set on the basis ofwhich an MLP may be trained and generated specifically for the cell typeof the cells used in the training cell cultures.

According to embodiments of the invention, the training data set isgenerated such that at each of a plurality of training points in timeduring the cultivation of multiple training cell cultures of cells ofthe specific cell type, the measured values and time specifications arereceived and the extracellular fluxes of the extracellular metabolitesare calculated. For this purpose the cell cultures are preferablycultivated in bioreactors of different types. Preferably, thesebioreactor types comprise at least two different types of bioreactorsfrom the following set: a fed-batch bioreactor, a batch bioreactor, aperfusion reactor (including variants with cell retention), a chemostatand a split-batch bioreactor.

The use of bioreactors of different types in the generation of thetraining data set may be advantageous, as a broader data basis isgenerated and an “overfitting” of the MLP in the course of training canbe avoided or reduced. In addition, it enables the use of the same MLPfor the successful prediction of the future metabolic state of a cellculture in many different types of bioreactors. Preferably, the trainingdata set is collected based on training cell cultures cultivated indifferent bioreactor types, whereby the bioreactor types include atleast one fed-batch bioreactor and/or at least one perfusion reactor.This may be advantageous, since these reactor types are being used moreand more frequently in practice and the representation of the metabolicstate of cell cultures in these reactor types has so far beenparticularly difficult due to their great dynamics.

Preferably, the cell culture whose future metabolic state is to bepredicted is cultivated in a type of bioreactor that was also used togenerate the training data sets.

According to embodiments, the training data set is generated in such away that at each of a plurality of training points in time during thecultivation of several training cell cultures of cells of the specificcell type, the measured values and time specifications are received andthe extracellular fluxes of the extracellular metabolites arecalculated, the cell cultures being cultivated in bioreactors of thesame type or of different types, all bioreactors not belonging to thebatch bioreactor type. For example, all bioreactors may be of thefed-batch type.

Due to the continuous or pulsed addition of additional culture mediumduring operation, the prediction of future metabolic states of cells infed-batch bioreactors has proven to be a very technical challenge.Embodiments of the invention are particularly advantageous in thecontext of the use of fed-batch bioreactors, since it has been shownthat predictions according to embodiments of the invention are accuratedespite the metabolic complexity of cell cultures cultivated in thistype of reactor.

According to some embodiments the MLP is a support vector machine or asystem of several support vector machines.

According to other embodiments, the MLP is a neural network or a systemof several neural networks (NNs).

Some initial tests suggest that other MLP methods may be used inaddition to support vector machines and neural networks. However,particularly good prediction results have been achieved when usingneural networks and a wide range of software solutions for differentnetwork architectures is already available that allow easy handling ofthe neural network during the training phase as well as during theapplication phase.

According to embodiments of the invention, the MLP is a system ofseveral sub-MLPs (in particular individual NNs), wherein the individualsub-MLPs contained in the system have each been trained to predict theextracellular flux of a single extracellular metabolite and areselectively used to predict the extracellular flux of that singleextracellular metabolite at the future point in time.

This may be advantageous, as it has been shown that the predictive powerof measured concentrations of extracellular metabolites is differentwith respect to individual extracellular fluxes of other extracellularmetabolites. The quality of the prediction may be improved by trainingindividual sub-MLPs, for example individual neural networks, each basedon a specific set of input parameter values with respect to theextracellular flux of the extracellular metabolite as an outputparameter value. The results of the individual sub-MLPs may be linked bya higher-level MLP or other program logic so that an extracellular flowof one or more extracellular metabolites is returned as an overallresult. A “sub-MLP” is an MLP that is functionally linked to one or morefurther “sub-MLPs” in such a way that the output of this and the further“sub-MLPs” is combined, e.g. aggregated, by a higher-level programlogic, in particular by a further, higher-level MLP, to form an overallresult.

According to embodiments of the invention, the MLP uses measuredconcentrations of several extracellular metabolites as input parametervalues to predict the extracellular flux of a single one of theextracellular metabolites. In this respect, the multiple extracellularmetabolites used as input parameter values for at least two of theextracellular metabolites whose extracellular flux is to be determinedare different.

This may be advantageous, as it may lead to a higher accuracy of theprediction.

According to embodiments of the invention, the method further comprisesa measurement of the concentrations of all input candidate metabolitesover several points in time. In particular, the measurement may serve toestablish metabolic concentration profiles over time. The set of inputcandidate metabolites comprises all extracellular metabolites which aremetrologically available in a reference bioreactor with a cell cultureof the specific type or all extracellular metabolites of the metabolicmodel. For example, the reference reactor may be one or more trainingbioreactors, i.e. reactors that were used to collect the training datafor the generation of MLP. Alternatively, the reference reactor may alsobe another reactor with a cell culture of the same type as thebioreactor currently being monitored. The determination of thoseextracellular metabolites that are to serve as input parameter valuesfor the MLP with regard to the prediction of the extracellular fluxes ofindividual extracellular metabolites of the model is thereforepreferably done during or before the training of the MLP.

For each of the extracellular metabolites whose extracellular flux is tobe predicted, a selection procedure is performed to identify themultiple extracellular metabolites to be used as input parameter valuesto predict the extracellular flux of that single metabolite. Theselection procedure comprises with respect to this single metabolite ineach case:

-   -   (a) defining a first set of extracellular metabolites, the first        set comprising all candidate input metabolites;    -   (b) calculating a first relevance score of each of the        extracellular metabolites in the first set as a function of the        measured concentrations of that metabolite, the first relevance        score indicating the predictive power of the concentration of        the respective extracellular metabolite with respect to the        extracellular flux of that single extracellular metabolite;    -   (c) transferring only that one of the extracellular metabolites        having the highest first relevance score from the first set to a        still empty second set of extracellular metabolites, removing        this metabolite from the first set;    -   d) calculating a further relevance score of each of the        extracellular metabolites in the first set as a function of the        measured concentrations of that metabolite and the measured        concentrations of all extracellular metabolites contained in the        second set, the further relevance score indicating the        predictive power of the concentration of the respective        candidate input metabolite with respect to the extracellular        flux of that single extracellular metabolite taking into account        the metabolites already contained in the second set;    -   (e) transferring only that one of the extracellular metabolites        of the first set which has the highest further relevance score        to the second set, removing this metabolite from the first set,        the transfer taking place only if, by the inclusion of this        metabolite, the second set does not exceed a maximum limit for        informative redundancy of the metabolites contained therein with        respect to the prediction of the extracellular flux of this        single extracellular metabolite;    -   (f) repeating steps d) and e) until no more metabolites can be        transferred from the first to the second set without the second        set exceeding the maximum informative redundancy limit; and    -   (g) using selectively only the metabolites transferred to the        second set as input parameter values to predict the        extracellular flux of that single extracellular metabolite.

According to embodiments, the first relevance score is calculated as apartial mutual information score—PMI score—between a metabolite of thefirst set and the single metabolite whose extracellular flow is to bepredicted. The second relevance score is calculated as a PMIscore—between a metabolite of the first set and the single metabolitewhose extracellular flow is to be predicted, taking into account allmetabolites already contained in the second set.

This may be advantageous, as it may allow the determination and use ofinput parameter values that have particularly high predictive power forthe respective extracellular metabolite or its flux. Overfitting in thecourse of training and thus poor prediction quality may be avoided.

By identifying those metabolites that have the highest significance(“relevance” or “predictive relevance”) for a specific extracellularflux of an extracellular metabolite, it is ensured that the selectedinput parameter values enable the generation of an MLP with goodpredictive power. Predictive relevance is preferably determined bydetermining the degree of correlation of a measured metaboliteconcentration profile of a specific extracellular metabolite with themetabolite concentration profile of the metabolite whose extracellularflux serves as the output parameter value of the MLP (i.e. whoseextracellular flux is to be predicted). Predictive relevance may bedetermined by various methods, e.g. principal component analysis or PMI(“partial mutual information”) as described below for embodiments of theinvention. The fact that a metabolite is only included in the second setif the second set does not already contain a metabolite whoseconcentration profile strongly correlates with the concentration profileof this metabolite (which implies a high degree of informativeredundancy of this metabolite with this metabolite already contained inthe second set) protects against the fact that the second set alsocontains groups of two or more metabolites whose concentration profilesstrongly correlate and would thus introduce redundant information intothe second set. A high proportion of information-redundant metabolitesin the second set would lead to overfitting effects.

For example, for the inclusion of further metabolites from the first tothe second set, the reaching of a maximum value for informativeredundancy or another termination criterion may be defined, so thatnormally only a part of the metabolites is transferred from the first tothe second set.

Thus, according to embodiments of the invention, the selection of theinput parameters (extracellular metabolites, whose concentration ismeasured and entered into the MLP) may be completely independent of theMLP.

However, it is also possible that the selection procedure is carried outin the form of a “wrapper” as described in chapter 4.7.3 of theappendix, e.g. as a functionality provided by the neural network.

According to embodiments, the first relevance score of each of themetabolites in the first set with respect to this single extracellularmetabolite is calculated as PMI score between this metabolite and thesingle metabolites whose extracellular flux is to be predicted in eachcase.

A PMI score (in contrast to the simpler Mutual Information (MI)) mayalso be used in the further, iteratively executed steps to calculate thefurther relevance score of each metabolite remaining in the first set.The further PMI score indicates the predictive relevance of thismetabolite with respect to the extracellular metabolite, whereby thisrelevance includes the informative redundancy of the metabolite to betested from the first set with respect to all metabolites alreadyincluded in the second set. Thus, before a metabolite is finallyincluded in the second set, it is checked whether it has any “predictiveadded value” over and above the metabolites already contained in thesecond set in view of the metabolites already in the second set. If themeasured concentration profile of this metabolite correlates stronglywith a metabolite already present in the second set, this is negated. Inthis case, there will be no inclusion in the second set.

A test data set may be used to estimate the quality of the selection ofinput parameters and to optimize the architecture of the MLP (e.g.number of layers of a neural network).

The determination of the predictive relevance of a metabolite withrespect to an output parameter metabolite by means of the PMI thusallows a measurement or estimation of the possible input variable andthe dependency between each of the possible input variables with respectto the output variable. The stronger the dependency, the better theoutput variable may be predicted on the basis of the input variables,the higher the relevance score of the input value metabolite. Thisenables the calculation of the relevance score and score-based sortingof the metabolites of the first set.

In the PMI-based decision whether a metabolite is included in the secondset, the PMI is used to determine the dependence of this metabolite oneach metabolite already in the second set. Only those metabolites areincluded in the second set that contain enough relevant new informationcompared to the metabolites already in the second set to avoidredundancies.

Thus, when selecting the next relevant metabolite in the second set, themetabolites already selected in the second set are also taken intoaccount.

The calculation of the dependency between extracellular metabolites inthe form of PMI (“partial mutual information”) may, for example, becarried out in order, in the course of the decision whether a metaboliteof the first quantity should be included in the second quantity, tocompare the calculated PMI with a PMI criterion, e.g. a PMI limit valuefor a still acceptable degree of dependence; however, the comparisonwith a PMI criterion usually does not consist of a simple comparisonwith a limit value, but consists of a statistical test or an alternativeselection procedure such as a “wrapper”, as explained in the appendix,for example, according to equations 2.22-2.27 and in chapter 4.7.3 ofthe appendix. According to the embodiment described in the Appendix, the“most relevant”, i.e. the one with the highest PMI value, is alwaystaken over into the first set. In this way an order of all metabolitesis generated. Finally, however, ALL metabolites are in the first set. Ina subsequent step (wrapper), it is then decided how many of the mostrelevant metabolites will be included in the “second list” of those usedfor prediction.

According to embodiments, the MLP uses measured concentrations ofseveral extracellular metabolites as input parameter values to predictthe extracellular flux of each of the extracellular metabolites, whereinthe several extracellular metabolites comprise at least one, preferablyat least two amino acids. The metabolites whose concentrations are usedas input parameter values often, but not necessarily, contain themetabolite whose flux is used as the output parameter value.

The use of amino acid concentrations in the medium as input parametervalues for the MLP or for the training of the MLP may be advantageous,since established methods for determining their concentration alreadyexist and the sufficient presence of amino acids is often necessary forthe efficient synthesis of many target proteins in a bioreactor.

It is possible that already during the training only the concentrationsof this subset of metabolites are selectively used to train the MLP.

If one sorts the input parameter values of the first set according totheir relevance (e.g. according to PMI), then in the end all availableinputs are contained in this “set”, but in the order that indicatesrelevance. An additional criterion is then used to determine from thislist the number of input parameters (concentrations) that are to be usedas inputs in the future. According to embodiments of the invention, theselection was made during the training itself (different numbers ofinput parameters were compared in terms of their predictive power, whichwas evaluated using a test data set).

Preferably, only the concentrations of this subset of metabolitesmeasured in the cell culture, whose metabolic status is to be predicted,are selectively used as input parameter values for the already trainedMLP. The selective use of predictively relevant and independent inputvariables (instead of all metabolically available concentrations ofextracellular metabolites) may be advantageous as it reduces the problemof “overfitting”, simplifies data collection (it may not be necessary tomeasure all extracellular metabolite concentrations of the model), andreduces the need for computer resources for prediction as fewer inputparameter values need to be evaluated.

According to embodiments, the metabolic model for the intracellularmetabolites of the model represents a steady state assumption that theamount of intracellular metabolites remains constant so that the sum ofthe incoming fluxes for each intracellular metabolite is equal to thesum of the outgoing fluxes of that metabolite.

According to embodiments, the plurality of points in time (used topredict the extracellular fluxes of the currently cultured cell culture)are separated by time intervals of 10 minutes to 48 hours, preferably1-24 hours. According to preferred embodiments, the time intervalbetween the points in time used for predicting extracellular fluxes isless than or equal to, or at most 20% greater than, the time intervalsused in the preparation of the training data set.

According to embodiments, the plurality of points in time are separatedby time intervals which are of equal length over the duration of thecell culture performance or whose length decreases towards the end ofthe cell culture performance. By preferred embodiments, the profile ofthe change in time intervals between points in time used to predictextracellular fluxes is identical or very similar to the profiles of thechanges in time intervals used in the preparation of the training dataset.

The specific cell type may be prokaryotic or eukaryotic.

In particular, the specific cell type may be a eukaryotic cell type.

It has been shown that despite the high complexity of eukaryoticmetabolic processes, embodiments of the invention are capable ofaccurately predicting the metabolic state of the cell, especially forfuture time periods ranging from hours to 1-2 days in the future.

For example, the specific cell type may be a mammalian cell type, e.g.HELA cells and others.

According to one embodiment, the specific cell type is Chinese HamsterOvary (CHO) cells.

According to embodiments, the specific cell type is a geneticallymodified cell type which is maintained and/or multiplied in a bioreactorfor the purpose of obtaining a biomolecule. For example, it may be agenetically modified cell line that expresses a specific protein, e.g.an enzyme or a specific antibody, and/or expresses it in particularlyhigh quantities.

After embodiments have been determined, the calculated intracellularfluxes are evaluated for plausibility and consistency and/or with regardto further quality criteria and, if necessary, modified by adding,removing or changing stoichiometric equations. The measured and/orpredicted extracellular metabolite fluxes are then transferred to themodified metabolic model and the intracellular fluxes are recalculatedand re-evaluated for plausibility and/or consistency. Thus the qualityof the metabolic model may be improved and, if necessary, adapted tospecific cell types or cell clones. Using these plausibility criteria,plausible reference values for the intracellular fluxes may also beobtained according to embodiments of the invention on the basis ofseveral experimental tests.

According to embodiments, the calculation of the intracellular flux ofone or more intracellular metabolites at each of the future points intime involves a calculation of several or preferably all intracellularfluxes of the metabolic model. Preferably, all intracellular fluxes ofthe model are calculated. The more intracellular fluxes are considered,e.g. in a plausibility estimate, the higher the reliability of theprediction.

According to embodiments, the method comprises an identification of allintracellular fluxes that deviate from a respective reference value orreference value range by more than the limit value. The reference valuesor reference value ranges may, for example, be obtained empiricallyand/or derived from the literature. The method further comprises anautomatic identification of that intracellular flux which acts as alimiting factor for cell growth or the production of a desiredbiomolecule.

For many metabolites their approximate intracellular flux is known inthe context of a specific metabolic state, e.g. by kinetic models, by13C-labelled substrates and quantification via NMR of isotopomers of themetabolites or by the amino acid composition of the cell proteins etc.Strong deviations from these reference ranges thus indicate that themetabolism of the cells in the cell culture is in an unfavorable or atleast unexpected state. The fact that embodiments of the inventioncompare intracellular fluxes with reference values and not, for example,the concentration of extracellular metabolites with reference values inorder to draw conclusions about the metabolic state of a cell via thesereference values, is advantageous, since this may allow a morefine-grained and better determination of (mostly undesired) deviationsof the physiological state of a cell from physiologically usual orfavorable reference values.

In a further aspect, the invention relates to a method for monitoringand/or controlling a bioreactor which includes the cell culture of cellsof a specific cell type. The method comprises a calculation ofintracellular fluxes at a future point in time according to theembodiments and examples of the method for predicting the metabolicstate of cells described herein. The method further comprises acomparison of the predicted intracellular fluxes with reference valuesor reference value ranges for acceptable intracellular fluxes of therespective one or more intracellular metabolites.

The method may be used to monitor the bioreactor and may include theissuing of a warning, the warning being issued if a deviation of thecalculated intracellular flux from its respective reference value orreference value range exceeds a limit value. The warning may, forexample, be sent via a graphical user interface to a human and/or viaanother interface to machines or software programs that log thedeviations.

In addition or alternatively, the method may be used to control thebioreactor and may involve sending a control command to the bioreactor.The control command is sent to automatically initiate steps to changethe condition of the bioreactor or the medium it contains to reduce thedeviation. For example, the control command may go to a valve, pump orother actuator in the reactor and cause the addition of culture media,trace elements, oxygen, CO2, pH-regulating acids or bases or acorresponding throttling of the addition. In particular, the automaticsteps may involve a change in the quantity or composition of a culturemedium. The control command may therefore be given to a mixer of aculture medium or a throttling unit at a feed line of the culturemedium, for example, to change the amount of specific sugars, saltsand/or amino acids in the culture medium or to reduce or increase thefeed rate of the culture medium into the bioreactor, depending on thepredicted fluxes.

For example, according to the embodiments of the method, it may bedetermined or predicted that a specific intracellular metabolic pathwayis significantly weaker (indicated by low intracellular flux) thanexpected or desired. It may be known that this metabolic pathway isoften limited by the amount of a specific vitamin or trace element inthe medium, e.g. iron. Therefore, if it is detected that this specificintracellular flux is lower than expected, the targeted addition of ironto the bioreactor may be counteracted much more specifically than ispossible if care is taken only to keep physical or extracellularparameters such as temperature, pH, glucose concentration etc. constant.

According to embodiments, the method for monitoring and/or controlling abioreactor involves identifying the reaction within the metabolic modelof the cells that acts as a limiting factor for cell growth or theproduction of a desired biomolecule according to the embodiments andexamples of the method for predicting the metabolic state and severalintracellular fluxes of cells described here. The method furthercomprises an automatic addition of selectively those substances(especially enzymes, co-enzymes, trace elements or nutrients) which(exclusively or especially) alter the intracellular flux acting as alimiting factor in such a way that cell growth or the production of thebiomolecule or the quality of the biomolecule is promoted. In additionor alternatively, the method comprises an output of a request for suchaddition via a user interface.

This may be advantageous as it allows more detailed monitoring or theadoption of very specific control measures to control a bioreactor thanis possible with methods known in the state of the art which are basedon trying to keep only the operating parameters of the bioreactorconstant, including some parameters measured in the medium. For example,a high uptake and metabolism of amino acids does not necessarily meanthat the cell also utilizes the amino acids for the synthesis of thedesired target protein. Depending on the state of the intracellularfluxes of the cell, it may also be that the absorbed amino acids aremetabolized for completely different purposes. However, according to theinvention, this may be achieved by metabolic flux analysis based on aspecific metabolic model of the cell using the predictions of the MLP.

After embodiments of the method, the predicted extracellular andintracellular fluxes are also used to test the quality of the model. Forexample, if a model formulated for a specific CHO cell clone differssignificantly from another clone or from another cell line (other tissueor animal species), the predicted fluxes (intracellular by MFA,extracellular by MLP) may be compared with measured rates ofconcentration change or plausibility criteria (no unrealistic orunphysiologically high fluxes, etc.). In case of high deviations of thepredicted from the measured or plausible fluxes, the model is adjustedor regarded as an indication for the presence of an error in themetabolic model on the basis of which the model is corrected.

By comparing the predicted fluxes with the measured or plausible fluxes,measurement errors in the determination of concentrations ofextracellular metabolites or biomass are identified according toembodiments, e.g. by means of statistical tests.

In a further aspect, the invention relates to a method for identifying ametabolically advantageous clone of cells of a specific cell type. Themethod comprises:

-   -   culturing of different cell cultures in several bioreactors,        whereby the different cell cultures contain different clones of        cells of the specific cell type;    -   calculating the intracellular flux of one or more intracellular        metabolites at several points in time separately for each of the        cell clones according to the embodiments and examples of the        method for predicting the metabolic state of cells described        herein;    -   identifying that one of the cell clones whose calculated        intracellular flux of the one or more intracellular metabolites        is metabolically most favorable.

This may be advantageous, since in the context of pharmaceuticalsynthesis processes it is often necessary to identify and selectivelypropagate metabolically advantageous cell clones. Many known methods forthe genetic modification of cells do not provide complete control overwhether and at which position in the genome of a cell a specific geneencoding a target protein to be synthesized is inserted. Depending onits position in the genome, the expression rate may vary. For example,the transfection of cells with viruses is a method in which manydifferent cell clones are created, some of which do not contain thedesired gene at all and others which have the desired gene inserted butat different positions in the genome. According to the invention, theparallel operation of several bioreactors with the different cell clonesin real time may now reveal whether the intracellular fluxes indicatethat the gene encoding the target protein has been incorporated into thegenome of the cell and that the target protein is synthesised in thecell to a considerable extent. For example, a comparison of the aminoacid composition of the target protein with the intracellular fluxes forthe synthesis or degradation of individual amino acids may give anindication as to whether the target protein has been incorporated. Inaddition or alternatively, the intracellular fluxes may provideinformation as to whether a specific cell clone reproduces sufficientlyfast and is vital, whether it has a low formation rate of toxic orotherwise undesirable metabolites, etc.

However, embodiments of the method according to the invention may notonly be used to predict the future metabolic state of cells, but also todescribe the current metabolic state of a cell.

According to embodiments, the method comprises:

-   -   calculating the current extracellular flux of one or more of the        extracellular metabolites from the concentrations of the        extracellular metabolites measured at the current point in time        and at the previous point in time;    -   performing a further metabolic flux analysis to calculate the        current intracellular fluxes at the current point in time using        the calculated current extracellular fluxes of the extracellular        metabolites and the stoichiometric equations of the metabolic        model; and    -   using the calculated current intracellular fluxes as a        description of a current metabolic state of the cells of the        cell culture.

This may be advantageous because it provides a very accurate assessmentof the current metabolic state of the cells in a cell culture down tothe level of individual intracellular fluxes.

According to embodiments the measurements also comprise a lactatedehydrogenase (LDH) concentration and at each point in time during thecultivation of the cell culture a LDH concentration measured in themedium of the cell culture is received. The predictions of theextracellular fluxes of the extracellular metabolites at each of thefuture points in time are made by the MLP using a corrected instead ofthe measured cell density.

Preferably, the calculation of the corrected cell density for each ofthe points in time comprises a calculation of the density of lysed cellsin the medium of the cell culture as a function of the measured LDHconcentration. This function may in particular be an empiricallydetermined heuristic and linear function representing the dependence ofthe LDH concentration in the medium on the number of lysed cells of thatspecific cell type. The corrected cell density is then calculated as thesum of the measured cell density in the medium and the calculateddensity of the lysed cells.

This may be advantageous because cells that are completely or partiallylysed are often not or only poorly detectable with optical methods fordetermining cell density, but the lysed cells may have had an influenceon the concentration of extracellular metabolites until shortly beforetheir lysis. Current methods for determining cell density only detectcells whose structure is still intact. This leads to a falsification ofthe total cell density determination if cell decay occurs, which tendsto happen at later points in time in fermentation. It was observed thatin some cases a systematic error was observed in the prediction of thefluxes with the method described here by embodiments, so that thepredicted fluxes were not in agreement with the measurable fluxes butrather showed a systematic error. It was observed that this errorcorrelated with the LDH concentration in the medium, which is anindicator for the presence of lysed cells in the medium. This enzyme isnot released into the medium by an intact cell. The detection of LDH inthe reaction medium thus indicates destroyed cells.

With the exception of batch fermentation, a similar, approximatelylinear relationship between the LDH concentration and the number oflysed cells has been demonstrated in many fermentation approaches.According to embodiments, the LDH concentration in the medium is used asa further measured value to calculate a corrected cell density.

According to embodiments of the invention, the corrected cell densitiesare used as input parameter values during training or application of thetrained MLP.

According to embodiments, in the MLP-based predictions of theextracellular fluxes, in addition to the concentrations of theextracellular metabolites, the LDH concentration is used as an inputvariable to predict at least some of the extracellular fluxes of themodel. According to embodiments, the LDH corrected value of cell densityis used as an output variable in addition or alternatively to theconcentrations of extracellular metabolites. Accordingly, according toembodiments of the invention, cell densities corrected by measured LDHconcentration are used in the MLP training to calculate extracellularfluxes and/or an LDH concentration is used as a further input parametervalue or as a concentration of an extracellular metabolite.

In another aspect, the invention relates to a system for predicting themetabolic state of a cell culture of cells of a specific cell type. Thesystem comprises one or more processors, a first interface for receivingmeasurements from a bioreactor containing the cell culture and avolatile or non-volatile storage medium.

The storage medium includes a metabolic model of a cell of the specificcell type, the metabolic model including a plurality of intracellularand extracellular metabolites and a plurality of intracellular andextracellular fluxes, the metabolic model comprising stoichiometricequations specifying at least one stoichiometric relationship betweenone of the intracellular and one of the extracellular metabolites. Thestorage medium further includes trained machine learning program logic(MLP) and program logic adapted to perform a method for predicting themetabolic state of the cells at each of a plurality of points in timeduring the cultivation of the cell culture. This method comprises:

-   -   receiving a plurality of measurement values measured at that        point in time via the first interface, said measurement values        comprising concentrations of a plurality of extracellular        metabolites of the metabolic model in the culture medium of the        cell culture and a measured cell density of the cells in the        cell culture;    -   inputting the received measured values as input parameter values        to the MLP;    -   predictions of extracellular fluxes of said extracellular        metabolites at a future point in time by said MLP using said        received measurement values, said future point in time being a        point in time subsequent to the time of receipt of said        measurement values, wherein said extracellular fluxes are uptake        rates of said extracellular metabolites into a cell and/or        release rates of said extracellular metabolites from a cell into        said medium;    -   performing metabolic flux analysis to calculate the        intracellular fluxes at the future point in time using the        predicted extracellular fluxes and the stoichiometric equations        of the metabolic model.

The metabolic model may, for example, be stored in the storage medium inthe form of a set of stoichiometric reaction equations defined inMatLab. The storage medium may be the main memory of a computer or anelectromagnetic or optical storage medium, e.g. a hard disk. The storagemedium may also be a distributed system of several hardware storageunits, for example a cloud storage area offered by a cloud service orthe IT infrastructure of a laboratory. The system may, for example, beimplemented as a standard computer, or as a notebook or portable mobiledevice of a user. The system may be integrated into a LaboratoryInformation System (LIS) or operationally connected to it. However, thesystem may also be a control computer for one or more bioreactors or acontrol module that is reversibly or irreversibly coupled to a singlebioreactor, for example as an integral part of the bioreactor.

For example, the first interface may be a network interface thatestablishes a wireless or wire-based connection to one or more sensorsof the bioreactor. In addition or alternatively, the first interface mayalso be an interface for manual input of the corresponding measuredvalues. For example, it is possible that the measured values areobtained by taking a sample from the bioreactor at specific points intime in a manual, semi-automatic or fully automatic method, which isthen transported to one or more further analysis devices, where the celldensity and/or the concentration of extracellular metabolites is thendetermined. The measured values obtained in this way may in turn beautomatically transferred from these analyzing devices to the system viathe first interface or a user may enter the measured values manuallyinto the system via a first interface designed as a graphical userinterface.

According to embodiments, the system is a control unit for monitoringand/or control of one or more bioreactors or is operatively linked tosuch a control unit. Preferably, the system further comprises a userinterface for outputting the calculated intracellular flux to a user.For example, the predicted flows may be displayed in tabular form or theflows may be dynamically visualized by means of a dynamic image, e.g. adynamic image based on a graphical representation of the metabolic modelunderlying the prediction of the intracellular flows.

According to embodiments of the system, the system also comprises asecond interface for sending control commands to the bioreactor. Thisprogram logic is adapted to:

-   -   comparing the predicted intracellular flux with reference values        or reference value ranges for an acceptable intracellular flux        of the respective intracellular metabolite(s);    -   issuing a warning via the user interface if a deviation of the        calculated intracellular flux of at least one of the        intracellular metabolites from its respective reference value or        reference value range exceeds a limit value; and/or    -   sending a control command to the bioreactor via the second        interface, the control command being adapted to change the state        of the bioreactor or the medium contained therein in such a way        that the deviation is reduced.

The user interface may, for example, be a graphical user interface (GUI)and/or an acoustic user interface. For example, a warning tone may begiven via the acoustic user interface or a warning in the form of atext, preferably with a qualified indication of the intracellular fluxfor which the deviation has been detected, may be displayed on a screenvia the GUI. The screen may be a screen coupled to the bioreactor or thescreen of a computer connected to one or more bioreactors via a network,for example a desktop computer, a server or a mobile communicationdevice, for example a user's smartphone.

The second interface may, for example, be physically adapted as awireless or wire-based connection between the system and the bioreactoror bioreactor actuators. The actuators may be pumps, valves for variousnutrients, buffers, pH-regulating liquids, trace elements, gases and/orstirrers or temperature controllers.

A “training point in time” is a point in time during the generation ofthe training record. In contrast, the “point in time” according to claim1 refers to a later point in time when the MLP trained on the trainingdata set is applied to predict the metabolic state of a currentlymonitored and/or controlled bioreactor.

A “metabolite” in the narrow sense of the term is an intermediateproduct (intermediate) in a biochemical metabolic pathway. However, ametabolite in the sense of the present invention shall be broader thanany substance which, in the form of an educt, product or intermediate,is involved in a biochemical reaction of a cell. In particular, ametabolite may be an amino acid, a sugar, fats, peptides, antibodies,proteins, components of the citrate cycle, components of glycolysis,components of protein synthesis or degradation pathways and similarsubstances.

An “extracellular metabolite” is understood here to be a metabolite thatis known or assumed to occur in the medium of a cell culture accordingto a metabolic model of the cell type under investigation, e.g. becauseit is secreted into the medium of the cell culture by cells of thespecific cell type (e.g. lactate) or because it is added to the cellculture as a component of the medium or a culture medium (e.g. glucose).

An “intracellular metabolite” is defined here as a metabolite that isknown or assumed, according to a metabolic model of the cell type underinvestigation, to occur within cells of the specific cell type, e.g.because it is taken up from the medium of the cell culture or isproduced by the cells.

A “flux” is a quantity of a substance that changes per time in aspecific volume due to a specific transport or reaction process. A fluxis therefore also called a “reaction rate” or “transport rate”. Ifseveral processes take place simultaneously in a specific volume, it ispossible that the net concentration of a substance does not change dueto the counter-rotational nature of some processes. e.g. throughsubstance conversion, uptake or release.

According to embodiments, the volume to which a flux indication refersis the volume of a cell for intracellular as well as extracellularmetabolites.

Because the change in the amount of substance caused by the specificprocess is related to the cell or the cell volume, a flux implicitlyalso indicates a change in the concentration of this substance per timein this volume caused by the specific process, which is caused by thisspecific reaction or transport process. Since, for a given volume, theflux of a substance implicitly also indicates a change in concentrationof this substance in the volume caused by the specific process and viceversa, the concept of a flux here should equally comprise a change in aquantity of substance per time in a volume as well as a quantity ofconcentration per time.

Preferably, the metabolic model used after embodiments of the inventionassumes that the amount or concentration of intracellular metabolitesremains approximately constant. However, this does not mean that theintracellular fluxes remain constant. Rather, a cell in whose cytosolthe concentration of an intracellular metabolite increases rapidly maycompensate for this, e.g. by increasing the reaction rate of one or morechemical reactions which metabolize this metabolite.

An “extracellular flux” is understood here to be the rate at which anextracellular metabolite is taken up or released by the cell via aspecific transport process. More precisely, it is the amount of themetabolite that is taken up into the cell per cell and per time by thisspecific transport process, or is released by the cell into thesurrounding medium. According to the embodiments of the invention, theextracellular flux v of a component at time t is determined on the basisof the measured change in the concentration of the correspondingmetabolite in the reactor medium, which is normalized with the livingcell density.

According to embodiments, for example, the change in concentration ofthe extracellular metabolite in the culture medium is measured over atime interval. This may be used to calculate the absolute change in theamount of metabolite in the medium (using the volume of the medium inthe reactor). The live cell count in the medium may be measured. Bynormalizing the absolute change in the amount of the extracellularmetabolite to the cell number, the extracellular flux may be given as aspecific (biomass related) quantity. Since the average volume of asingle cell of the cultured cell type is usually known from literatureor may be measured, a conversion of the measured change in concentrationto the average cell volume is performed according to embodiments of theinvention.

For example, the metabolic model of cell metabolism used for metabolicflux analysis (also “flow analysis” or “material flow analysis”) mayinclude extracellular stoichiometric reaction equations, each of whichis associated with an extracellular flux. Thus, extracellular fluxesdescribe in particular uptake rates of extracellular metabolites intothe cells of a cell culture, so that these then function asintracellular metabolites, and release rates of extracellularmetabolites of the cells of the cell culture to the surrounding medium,so that these then occur there as extracellular metabolites.

Via these extracellular fluxes, which represent transport fluxes intoand out of the cell, the purely intracellular fluxes are linked to theconcentrations and concentration changes of the extracellularmetabolites in the metabolic model and allow the determination ofplausibility criteria based on the stoichiometries specified in themodel and the empirically determined or allow descriptive assessments ofcurrent intracellular fluxes of the cell and/or predictions of futureintracellular fluxes of the cell on the basis of the extracellularfluxes using the stoichiometries, plausibility criteria and theempirically determined or predicted concentration changes of theextracellular metabolites in the medium specified in the model.

The intracellular and extracellular fluxes are thus coupled together inthe metabolic model via one or more intracellular metabolites, sincesome substances occur both as intracellular metabolites, whose rate offormation from or metabolization into one or more intracellularmetabolites is described, and whose rate of import into the cell orrelease from the cell is described by extracellular fluxes of the model.

A change in the concentration of an extracellular metabolite and thus anextracellular flux may also be caused in some types of bioreactors by anexternal addition or feeding of this metabolite into the cell culturemedium. An extracellular flux may be calculated approximately, e.g. byfirst determining the absolute difference in the metaboliteconcentration measured at two consecutive points in time and thenconverting this difference to the measured cell density. The higher thecell density, the lower the flux per cell, since the absolute measuredchange in metabolite concentration is distributed over a larger numberof cells. The uptake flux of an extracellular metabolite at the futurepoint in time may therefore be calculated as the difference inmetabolite concentration measured in the time interval between the twopoints in time divided by the duration of the time interval, the resultthen being divided by the measured cell density.

An “intracellular flux” is understood here to be the rate (a quantityrelated to a time interval) at which a reaction takes place within acell, whereby the reaction may consist of: a transport betweenintracellular compartments (e.g. transport from the cytosol into themitochondria and vice versa) or a conversion of one or moreintracellular metabolites (educts) into one or more other metabolites(products) in the cell. In case of an intracellular flux, all educts andproducts are intracellular metabolites.

Intracellular fluxes are formulated in the metabolic model forstoichiometric equations that specify a reversible or irreversiblereaction of the above categories (intracellular transport between cellcompartments, metabolic transformation).

“Unmeasurable fluxes” are fluxes which may not be readily determinedfrom the fermentation data. In general, intracellular fluxes are notmeasurable because of the difficulty in observing intracellularmetabolites.

According to preferred embodiments of the invention, MFA for thedetermination of intracellular fluxes based on provided extracellularfluxes is based on the assumption that the concentration of anintracellular metabolite does not change. Concentrations ofintracellular metabolites are very difficult or impossible to measure.Embodiments of the invention are based in MFA on the assumption that thesum of input and output flows at an intracellular metabolite (whoseconcentration is unknown) is identical and therefore its concentrationdoes not change. Depending on the operating mode of the reactor, this isnot exactly correct, but at least approximately sufficiently correctwith regard to the time intervals described here, since the cellmetabolism returns to its chemical equilibrium quite quickly.

According to embodiments, the metabolic model used in MFA comprises atleast 10, preferably at least 20 stoichiometric equations. The model ispreferably adapted to describe metabolic processes in the cellcompletely or at least approximately completely.

The use of a metabolic model which comprises the cell metabolism ascomprehensively as possible may be advantageous, since for the selectionof specific cell clones it is important to have as complete a picture aspossible of the individual metabolic activities and particularities ofeach cell clone investigated. A pure plausibility check of theextracellular fluxes predicted by MLP by means of individualstoichiometric equations would generally not provide sufficient data tobe able to select specific cell clones based on this model due to theiradvantageous metabolic properties.

In another advantageous aspect, the use of the metabolic model with alarge number of stoichiometric equations in MFA may also be used tomonitor cell cultures and, in case of deviation of one or moremetabolite fluxes from a set point range, to make specific changes inthe amount and/or composition of the nutrient solution supplied or othercontrol parameters (temperature, pH, partial pressure oxygen, partialpressure CO2, speed stirrer etc) automatically and/or manually.

A “metabolic model” is understood here to be a descriptive model of thecurrent metabolic state of cells of a specific cell type. The metabolicmodel is preferably reduced in complexity compared to the real reactionstaking place in the cell and is restricted to those parts of the cellmetabolism which are of particular interest for the respectiveapplication. Preferably, the model includes several extracellular andintracellular metabolites, reaction and transport equations withstoichiometric factors, and extracellular and intracellular fluxes. Themetabolic state of a cell may be characterized, at least in part, byindicating the level of the individual intracellular fluxes at aspecific point in time. Preferably, the temporal change of metaboliteamounts in the cell results from the balancing of incoming and outgoingmaterial flows, as specified in the extracellular and intracellularreaction equations of the model. The model is an MFA model suitable forperforming metabolic flux analysis and is based on the so-calledsteady-state assumption that the amount of intracellular metabolitesremains constant, which means that the sum of the incoming fluxes foreach intracellular metabolite is equal to the sum of the outgoing ones.

The concentrations of extracellular metabolites measured and used asinput parameter values of a trained MLP are, according to embodiments ofthe invention, metabolite concentrations in the strict sense. Ametabolite concentration in the narrow sense of the term is understoodhere to be a content specification related to a volume. Theconcentration thus indicates how much of a metabolite is present in areference volume (e.g. cell culture medium). The metaboliteconcentration may, for example, be indicated as mass concentration inthe unit g/l or as substance quantity concentration in the unit mol/l.

According to other embodiments, the concentrations of extracellularmetabolites measured and used as input parameter values of a trained MLPare metabolite concentrations in the broad sense. A metaboliteconcentration in the broader sense here means measured values and valuesderived therefrom which are known to correlate with a metaboliteconcentration in the narrower sense of the respective metabolite in alinear manner or at least approximately (at least 90% in theconcentration range in question) in a linear manner. For example, anextracellular flux of an extracellular metabolite may be understood as ametabolite concentration in a broader sense. The measured extracellularflux refers to the change in the concentration of the metabolite in themedium in the period between a point in time in the past, e.g. the lastmeasurement of the metabolite concentration, and a current point in timeat which the measured values are currently collected and used as inputfor the MLP. The essentially linear relationship between measured fluxand extracellular concentration results from the fact that an increasein the extracellular flux of a metabolite by a specific amount causes acorresponding change in the concentration of the metabolite in theextracellular medium. Furthermore, according to embodiments of theinvention, the measured metabolite concentrations may still be modifiedin various ways by offsetting with correction and normalization factors,so that these modified values ultimately also represent metaboliteconcentrations in the broader sense, i.e. correlate in a linear mannerwith the originally measured values, but are not identical to them.Since the actually measured metabolite concentration and the metaboliteconcentration in the broader sense correlate linearly with each other orthe metabolite concentration may be derived from the measured metaboliteconcentration in the narrower sense, both types of metaboliteconcentration data may ultimately be used equally as input for an MLP.

A “descriptive model” is a model that describes the current static anddynamic metabolic state of a cell.

A “predictive model” is a model that allows the prediction of a futurestatic and dynamic metabolic state of a cell. By embodiment, themetabolic model used for flux analysis is a descriptive metabolic model.

A “metabolic hybrid model” is a combination of a mechanistic model,which represents reaction kinetic knowledge, and an empirical model,which relates measured values from a bioreactor to metabolic states of acell.

A “metabolic flux analysis” (MFA) is a computational method that may beused to estimate intracellular fluxes from extracellular fluxes. Theextracellular material flux describes the amount of material that isabsorbed or released by a cell over time. The intracellular materialflux describes the reaction rate with which an intracellular metaboliteis converted or formed. MFA comprises different approaches to determinethe rate of metabolic reactions within a biological unit. Metabolism isa dynamic process and may be regarded as a kind of “cellular phenotype”described by MFA.

A “batch bioreactor” is a bioreactor which is operated in a “batchprocess” or is adapted to operate in a batch process. The batch methodis characterized by the fact that all substrates (especially sugar andamino acids) are presented in the bioreactor before being inoculated.Gases and PH correction agents, on the other hand, are also introducedinto the system during the process. Approximately, however, the batchprocess may be regarded as a closed system with a constant reactionvolume. Due to the gradual consumption of the substrates by the cells,an initial unlimited growth phase (so-called exponential phase) isfollowed by a phase of stagnation (stationary phase) in which the growthand death of the cells are in equilibrium. This is followed by the deathphase with a decrease in cell density as a result of a severe lack ofnutrients. Among the usual methods of bioreactor operation, the batchmethod is the least costly and least likely to cause contamination.However, the batch process does not usually result in an optimal productyield; the process duration is limited by the consumption of thesubstrates.

A “fed-batch bioreactor” is a bioreactor which is operated in an inletmethod (“fed-batch method”) or is adapted to operate in a fed-batchmethod. In the fed-batch method, additional culture medium is addedduring the process (often only after an initial batch phase). The feedmay be continuous or in the form of one or more highly concentratedboluses (i.e., pulsed). Compared to the batch method, the process timecan be extended, since used substrates may be re-dosed. Furthermore, abetter process control may be achieved. For example, inhibitionphenomena and the formation of toxic by-products can be contained bykeeping educt concentrations continuously lower. Overall, the fed-batchmethod may achieve significantly higher cell densities and productyields than the batch method.

A “chemostat” (“continuous reactor”) is a bioreactor to which a constantflow of culture medium is supplied and from which reaction mediumcontaining cells and products is withdrawn in equal measure. Thus thereaction volume does not change during fermentation. If the volume flowsare chosen appropriately, a steady state equilibrium is established inthe reactor, in which the cell density and the nutrient concentrationsremain constant.

A “perfusion bioreactor” or “perfusion reactor” is a bioreactor in whicha continuous stream of culture medium is added and a continuous streamof (usually cell-free) reaction medium is withdrawn. A continuous streamof cell-containing reaction medium may also be removed by means ofdirected “bleeding”. The choice of volume flows is linked to one or moreprocess control parameters and adjusted so that the reaction volume inthe reactor remains constant. Cell density and nutrient concentrationsmay be constant (related to the cell density) or variable (similar tofed-batch operation) depending on the process design.

A “split-batch bioreactor” is a bioreactor of the batch or fed-batchtype operated in such a way that a substantial part of its medium, e.g.more than 10% or more than 30%, has been removed from the reactor one ormore times for the purpose of harvesting the cells contained therein.

The “PMI” (partial mutual information) is a data value which quantifiesnon-linear direct dependencies of two parameters. In the context of somemachine learning approaches, e.g. neural networks, the PMI is a measurefor the dependence between a random input variable X and a random outputvariable Y, taking into account already selected inputs. Differentapproaches to calculate the PMI of two variables are known, e.g. SharmaA (2000): “Seasonal to interannual rainfall probabilistic forecasts forimproved water supply management: Part 1—A strategy for system predictoridentification”, Journal of Hydrology Vol. 239, Issues 1-4, 232-239.

A “PMI criterion” thus refers to a feature or characteristic that hasbeen established with regard to PMI in order to make a decision. Forexample, a PMI criterion may be a limit value, the exceeding or fallingbelow of which influences the course of a method.

Some examples of the invention are explained in greater detail in theappendix attached to this application, the disclosure content of whichis part of this application. In order to ensure consistency between thedescription and the appendix, the meaning of the variables as specifiedin the variable directory of the appendix has been retained. With regardto the meaning of the variables, reference is made to the list ofvariables in the appendix. Examples and embodiments as well as furtherexplanations described in the Appendix can be freely combined with theembodiments, examples and features described in the application text,provided they are not mutually exclusive.

SHORT DESCRIPTION OF THE FIGURES

In the following, embodiments of the invention are described in moredetail in an exemplary manner, whereby reference is made to the figureswhich each represent embodiments of the invention or individual aspectsof these embodiments.

FIG. 1 shows a flow chart of a method for predicting the metabolic stateof a cell;

FIG. 2 shows an example of the process of obtaining information inseveral stages using different devices and data sources;

FIGS. 3A and 3B shows a block diagram of a system for predicting themetabolic state of a cell, which may be used to monitor and/or controlone or more bioreactors;

FIGS. 4a and 4b shows a metabolic model of a cell with multipleintracellular and extracellular fluxes of intracellular andextracellular metabolites;

FIG. 5 shows the calculation of intracellular fluxes at severalconsecutive points in time during the operation of a bioreactor;

FIG. 6 shows fluxes of different metabolites according to the metabolicmodel shown in FIG. 4;

FIG. 7 shows several metabolite flows illustrating the successful use ofthe method for generating biological knowledge;

FIG. 8 shows several metabolite flows illustrating the successful use ofthe method for generating biological knowledge;

FIG. 9 shows plots with lactate fluxes and glutamine concentrations;

FIG. 10 shows time courses of intracellular fluxes;

FIGS. 11A-11D shows several intracellular and extracellular fluxes atdifferent points in time during the cultivation of a cell culture;

FIG. 12 shows the strongly correlated course of intracellular fluxes,which were calculated for the current point in time by descriptive MFAand predicted for a point in time in the future by a combination of theMLP and MFA;

FIG. 13 shows input parameter values and output parameter values of anNN;

FIG. 14 shows a histogram of the obtained RMSE for intracellular fluxesin 12 fed-batch fermentation runs;

FIG. 15 shows a histogram of the obtained RMSE for extracellular fluxesin 12 fed-batch fermentation runs;

FIG. 16 shows 12 plots each with one predicted extracellular metaboliteflux and two extracellular metabolite fluxes measured for identical cellclones in two different bioreactors (fed-batch and split batch);

FIG. 17 shows 11 plots, each with two curves, all obtained for a fedbatch bioreactor using two different calculation methods;

FIGS. 18A and 18B shows 11 plots, each with two curves of calculatedextracellular fluxes of a cell clone ZK1;

FIGS. 19A and 19B shows 11 plots with two curves each of calculatedextracellular fluxes of a cell clone ZK2;

FIGS. 20A and 20B shows 11 plots with two curves each of calculatedintracellular fluxes of a cell clone ZK1; and

FIGS. 21A and 21B shows 11 plots with two curves each of calculatedintracellular fluxes of a cell clone ZK1.

FIG. 1 shows a flow chart of a method for predicting the metabolic stateof a cell culture of CHO cells according to an embodiment, which methodis equally suitable for other cell types.

Step 102: Model Generation

The processes in a bioreactor can be described mathematically. First ofall, the mapping of the temporal changes of relevant substanceconcentrations or quantities in the reaction medium should be considered(e.g. courses of substrate quantities, product quantities, celldensities). The formulation is based on mass balances and consists of aterm that describes the reaction of the substance and a convection termthat comprises any material flows into and out of the reactor. Itapplies in general (see [51], section 4.2 in the Appendix):

$\begin{matrix}\text{time-related~~change} \\\text{in~~the~~substance} \\\text{amount~~of~~a} \\\text{component~~in~~the} \\\text{system}\end{matrix} = {\underset{{convection}\mspace{14mu} {term}}{\underset{}{\begin{matrix}\begin{matrix}\text{component} \\\text{added~~to}\end{matrix} \\\text{the~~system} \\\text{over~~time}\end{matrix} - \begin{matrix}\text{amount~~of~~the} \\\text{component} \\\text{withdrawn~~from~~the} \\\text{system~~over~~time}\end{matrix}}} + \underset{{reaction}\mspace{14mu} {term}}{\underset{}{\begin{matrix}\text{amount~~of~~the} \\\text{component} \\\text{converted~~in~~the} \\\text{system~~over~~time}\end{matrix}}}}$

or in equations:

$\begin{matrix}{{\frac{dm}{dt} = {{{\overset{.}{V}}_{zu} \cdot c_{zu}} - {{\overset{.}{V}}_{ab} \cdot c_{ab}} + Q}},} & (2.1)\end{matrix}$

wherein m the amount of substance in the reaction medium, t the processtime, {dot over (V)}_(zu) or {dot over (V)}_(ab) the volume flow of theinlet or outlet, c_(zu) or c_(ab) the concentrations of thecorresponding substance in the inlet or outlet and Q the amount ofsubstance that is converted per time and volume. In the case thatquantities of extracellular metabolites are considered, the reactionterm primarily comprises the uptake or release of the substance by thecells. If the cell density in the fermenter is to be described, itincludes the formation and death of the cells.

According to embodiments, the metabolic model is based on the assumptionaccording to the above equation that the temporal course of the amountof substance is differentiable. This is justified provided that allincoming and outgoing fluxes are continuous. In the case of bolusfeeding or sampling during fermentation, the result is a continuouspiece-wise curve. The above equation then applies to the areas betweenthe discontinuities.

On this differential equation numerous mathematical models may bespecified for process description, control and optimization. These playan ever-increasing role due to the growing desire to better understandbioprocesses and improve them in silico while saving expensive andtime-consuming laboratory experiments. Models that are based only onsuch mass balances and do not describe intracellular processes are knownas black box models. They may not be able to explain dependenciesbetween the considered processes in a mechanistic way. To do so, itwould be necessary to model the metabolism as a link between thedifferent extracellular substances.

Ultimately, the metabolic model should enable a metabolic material flowanalysis to be carried out, so that the model may be used to drawconclusions from extracellular fluxes to intracellular fluxes. Whileeasy-to-use methods are generally established for determining celldensity and measuring extracellular substance concentrations, theobservation of intracellular reaction rates is much more complex. Toavoid such experiments, metabolic flux analysis (MFA) has beendeveloped—a computational method that may be used to estimateintracellular fluxes from extracellular fluxes. The extracellularmaterial flux describes the amount of material that is absorbed orreleased by a cell over time. The intracellular material flux is theamount of material that is converted in an intracellular reaction pertime and cell.

Thus, in order to be able to perform metabolic material flow analysis, a(preferably or usually simplified) biochemical, stoichiometric metabolicnetwork of the organism under consideration is generated first, whichcomprises the most important intra- and extracellular reactions.Extracellular reactions are—analogous to fluxes—those in whichmetabolites are taken up or released by the cell.

It is assumed that the network consists of k reactions of which k_(m)the results are measurable and therefore known (these are usually theextracellular reactions) and of l intracellular metabolites. Thereactions can then be recorded in a stoichiometric matrix A∈

^(k) in which the stoichiometric coefficients (negative for educts,positive for products of the individual reactions) are entered, with therows corresponding to the various metabolites and the columnscorresponding to the reactions. The extracellular metabolites areomitted. A concrete example is given in section 2 of the Appendix.

The material flux of the j-th reaction is designated with v_(j).Furthermore, m_(i) is the amount of the i-th intracellular metabolite ina single cell. If the fluxes and metabolite quantities are combined tovectors v or m, the following applies:

$\begin{matrix}{\frac{dm}{dt} = {Av}} & (2.2)\end{matrix}$

The equation states that the temporal change of metabolite quantities inthe cell results from the balancing of incoming and outgoing materialflows.

The MFA is then based on the so-called steady-state assumption that theamount of intracellular metabolites remains constant, which means thatthe sum of the incoming fluxes for each intracellular metabolite equalsthe sum of the outgoing ones. This simplifies equation (2.2) to

0=Av  (2.3)

Dividing the vector v into the sub-vector v_(m) ∈

^(k) ^(m) of known (measurable) and the sub-vector v_(u)∈

^(k−k) ^(m) of unknown fluxes and the matrix A correspondingly into thesubmatrices A_(m)∈

^(l×k) ^(m) and A_(u)∈

^(l×(k−k) ^(m) ⁾, so that Av=A_(m)v_(m)+A_(u)v_(u), equation (2.3)becomes

A _(u) v _(u) =−A _(m) v _(m).  (2.4)

The determination of unknown fluxes v_(u) by MFA may thus be carried outby solving a linear system of equations.

The next step is to classify the system of equations. The obviousformulation for the solution of the system of equations

v _(u) =−A _(u) ⁻¹ A _(m) v _(m)

is usually not applicable, since the matrix A_(u) is usually notinvertible. In such cases the solution space may be infinite orcontradictions may occur so that no solution exists. The following termswere introduced by van der Heijden et al. (1994), which classify thesystem of equations or material fluxes according to criteria ofsolubility and consistency [17, 39, 40], and which are also used for thegeneration of the model according to the embodiments of the invention:

Determination:

The system (2.4) is under-determined, if Rang(A_(u))<k−k_(m) isapplicable. In this case not all unknown fluxes may be calculatedunambiguously, because the metabolic network contains too fewrestrictions. If Rang(A_(u))=k−k_(m), then the system has at most onesolution and is called determined.

Redundancy: If Rang(A_(u))<l, the system is redundant. This means thatlinearly dependent lines exist in A_(u). Due to measurement errors inthe determination of v_(m) or inaccuracies in the metabolic networkmodel, this usually leads to an inconsistent system for which nosolution exists (it applies then Rang(A_(u))<Rang((A_(u)|−A_(m)v_(m))),the latter term represents the extended coefficient matrix). IfRang(A_(u))=l, the system is not redundant and therefore alwaysconsistent.

Calculability:

A flux v_(u) is called calculable if it may be unambiguously calculatedusing equation (2.4), otherwise it is not calculable. A consistentsystem is assumed here. If the system is under-determined, there is atleast one flux that is not calculable.

Balanceability:

A flux v_(m) is called balanceable if its value has an influence on theconsistency of the system, otherwise not balanceable. Balancable flowsonly occur in redundant systems.

As mentioned above, MFA systems are often under-determined and/orredundant (although a system may be under-determined and redundant atthe same time). A solution can then be formulated using theMoore-Penrose pseudo-inverse A_(u) ^(#), which is defined for all A_(u):

v _(u) =−A _(u) ^(#) A _(m) v _(m).  (2.5)

In the case of underdeterminedness, this expression yields one of theinfinitely many solutions to the system of equations; in the case ofinconsistency, it yields a least-squares solution.

In the following, methods will be presented with which it may be checkedwhich of the unknown fluxes are nevertheless calculable in the case ofunderdetermination, and which of the measured fluxes can be balanced inthe case of redundancy.

Identification of Calculable Fluxes

The method for the identification of calculable fluxes published byKlamt et al. is presented [17]. Be T∈

^((k−k) ^(m) ^()×(k−k) ^(m) ^(−Rang(A) ^(u) ⁾⁾ a matrix whose columnsform a base of the core of A_(u). Then it applies:

A _(u) T=0.

Each vector p∈Kern(A_(u)) can be represented as a linear combination ofthe base vectors. So there is a a∈

^(k−k) ^(m) ^(−Rang (A) ^(u) ⁾, so that applies

p=Ta.

Equation (2.4) can be extended:

A _(u) v _(u) =−A _(m) v _(m) +A _(u) Ta,

where a∈

^(k−k) ^(m) ^(−Rang (A) ^(u) ⁾ is arbitrary. The left-sided applicationof the pseudo-inverse then results in

v _(u) =A _(u) ^(#) A _(m) v _(m) +Ta.

By variation of the vector a one obtains the space of the solutions ofthe system of equations (2.4). τ has a rank greater than 0 exactly ifthe system is underdetermined. From this it can be concluded that thecalculable fluxes v_(u) are exactly those on which variation of a has noinfluence. This is exactly the case if the corresponding row in thematrix T is a zero row.

Preferably, according to the embodiments of the invention, the next stepis the identification of calculable fluxes.

For a non-redundant system, inserting (2.5) into (2.4) yields theformulation

Rv _(m)=0  (2.6)

with the redundancy matrix R: =A_(m)−A_(u)A_(u) ^(#)A_(m) (seereferences [17, 39] in the appendix).

However, if the system is redundant, equation (2.6) is only fulfilledfor determined ones, which is equivalent to the solvability of equation(2.4). The column notation

RV _(m) =r ₁ v _(m,1) + . . . +r _(k) _(m) v _(m,k) _(m) =0

illustrates that a measured flux has no influence on the solvability ofequation (2.4) if the corresponding column vector R is of the nullvector. r_(j) denotes in the above formulation the j-th column vector ofR and v_(m,j) the j-th measured flux.

In the following, possibilities are presented for treatingunderdetermined and redundant

Treatment of Under-Determined Systems

If the system is under-determined, i.e. has an infinite number ofsolutions, there are several ways to modify the problem in order toarrive at a unique solution. One option would be, if possible, totighten the resulting restrictions by extending the metabolic networkmodel, thus creating a determined system. Furthermore, one may try toincrease the number of known fluxes by additional experimentalquantifications. The determination of intracellular fluxes may beachieved by ¹³C-labelling experiments (see literature [43] in theappendix).

f one wants to avoid this experimental effort, the widespread method ofFlux Balance Analysis (FBA) is a good choice. Here, a target function tobe optimised is defined, which is selected according to biologicalplausibility and which depends on the material fluxes. For example, itcan be assumed that the host organisms direct their material fluxestowards maximising their growth rate, as this represents a significantevolutionary advantage. If the target function is designated F, then theFBA results in the general formulation:

max   F(v)   s.t.  Av = 0

By formulating it as an optimization problem with equation constraints,a clear flux distribution is usually provided as the solution. Ifinformation about irreversibilities of reactions is available, theallowable range (allowable range: set of points for which allconstraints of the optimization problem are fulfilled) can be furtherrestricted by the additional inequality conditions

v _(irrev)≥0

whereby v_(irrev) is the vector of all irreversible fluxes.

The main problem in FBA is the correct choice of the objective functionon which the solution depends. It is quite possible that cells changetheir biological target during fermentation (see literature [33] in theappendix).

Management of a Redundant System

In the case of a redundant system of equations, there is usually no fluxdistribution that solves equation (2.4) due to contradictions in theflows that can be balanced. Even in a completely correctly definednetwork model, inconsistencies usually occur, caused by measurementerrors in the determination of v_(m). The actually measured flow valuesshould be designated in the following as v _(m)=(v _(m,1), . . . , v_(m,k) _(m) )^(T) in order to clearly distinguish them from the truemeasurable values. In the literature there are several methods tocalculate an approximate solution {circumflex over (v)}=[{circumflexover (v)}_(m), {circumflex over (v)}_(u)] of a redundant MFA problem:

One possibility would be to set {circumflex over (v)}_(m)={circumflexover (v)}_(m) and select the vector {circumflex over (v)}_(u) so thatthe mean square distance between the vectors A_(u){circumflex over(v)}_(u), and −A_(m) v _(m), is minimized. The formulation asoptimization problem is then

$\begin{matrix}{\min\limits_{v_{u}}{{{{A_{u}v_{u}} + {A_{m}{\overset{\_}{v}}_{m}}}}^{2}.}} & (2.7)\end{matrix}$

The corresponding solution {circumflex over (v)}_(u)=−A_(u) ^(#)A_(m) v_(m) only approximately fulfils the steady-state assumption. If thesystem is determined, then A_(u) ^(#)=(A_(u) ^(T)A_(u))⁻¹A_(u) ^(T)applies.

A second possibility is the determination of a least-squares solutionfor the vector v, which minimizes the relative squares distance to themeasured fluxes (see literature [39] in the appendix) and fulfills thesteady-state condition. The formulation for this is:

$\begin{matrix}{{\min\limits_{v}\mspace{11mu} {\frac{1}{2}{\sum_{j = 1}^{k_{m}}\left( \frac{v_{m,j} - {\overset{¯}{v}}_{m,j}}{{\overset{¯}{v}}_{m,j}} \right)^{2}}}}{{s.t.\mspace{11mu} {Av}} = 0}} & (2.8)\end{matrix}$

Irreversibilities may also be included as additional constraints.

The second method differs from the first one mainly in that its solutionfulfils the steady-state assumption (the measured fluxes are then calledbalanced). For this, the values of the measurable fluxes in the solutiononly approximately correspond to the values actually measured. This maybe considered reasonable if the steady-state assumption is considered tobe more reliable than the measured values for the fluxes, which arealways subject to errors. In the solution of (2.8) all those measuredfluxes are adjusted which can be balanced. Those that cannot be balancedremain unchanged.

A generalization of the method just described for adjusting balanceablefluxes results from a more statistically motivated approach, which isexplained in the literature cited in the appendix [39, 40]. It is basedon a weighted least squares approach:

Be v _(m) again the vector of the measured, faulty fluxes and be v_(m)the vector of the corresponding true values. The measurement errorvector δ denotes the difference between the true and the measuredvalues:

δ:=v _(m) −v _(m).

It is assumed that δ is to o be normally distributed with expectationvector κ and covariance matrix

C _(v) _(m) : =

[δδ^(T)].

Since v_(m) is unknown, a plausible, experience-based estimate of thecovariance matrix must be used for further calculations.

The goal is now an estimate of v_(m), which fulfills the steady-stateassumption and at the same time is close to v _(m), taking into accountthe information about covariances. This is achieved by using theMahalanobis distance. The optimization problem for this is:

$\begin{matrix}{{\min\limits_{v_{m}}{\delta^{T}C_{{\overset{¯}{v}}_{m}}^{- 1}\delta}}{{s.t.\mspace{11mu} {Rv}_{m}} = 0}{\delta = {v_{m} - {\overset{¯}{v}}_{m}}}} & (2.9)\end{matrix}$

The solution {circumflex over (v)}_(m), of the optimization problem hasthe form

v _(m)=(I−C _(v) _(m) R′ ^(T)(R′C _(v) _(m) R′ ^(T))⁻¹ R′) v_(m)  (2.10)

(see literature reference [23] according to appendix). I is the unitmatrix. R′ is the reduced form of R, which is generated by eliminationof linear dependent lines and therefore has full rank (this is notunique). It may be generated by multiplication with a non-square matrixΓ, which carries out the corresponding line transformations:

R′=ΓR.  (2.11)

These balanced values may now be used in equation (2.5) to calculate theunknown fluxes.

Assuming that the measurements of the fluxes are independent (i.e. thecovariance matrix C _(v) _(m) is diagonal) and the standard deviation ofv_(m,j)−v _(m,j) is proportional to the magnitude of the measured valuev _(m,j) with a uniform proportional constant b>0, the optimizationproblem (2.9) is equivalent to (2.8). It applies:

${\delta^{T}C_{{\overset{¯}{v}}_{m}}^{- 1}\delta} = {{\left( {v_{m} - {\overset{¯}{v}}_{m}} \right)^{T}\frac{1}{b^{2}}\begin{pmatrix}\frac{1}{{\overset{\_}{v}}_{m,1}^{2}} & \; & 0 \\\; & \ddots & \; \\0 & \; & \frac{1}{{\overset{\_}{v}}_{m,k_{m}}^{2}}\end{pmatrix}\left( {v_{m} - {\overset{¯}{v}}_{m}} \right)} = {\frac{1}{b^{2}}{{\Sigma_{j = 1}^{k_{m}}\left( \frac{v_{m,j} - {\overset{¯}{v}}_{m,j}}{{\overset{¯}{v}}_{m,j}} \right)}^{2}.}}}$

The formulation (2.9) offers the advantage over (2.8) that thecovariance matrix may be flexibly adapted to the quality of the measureddata. When balancing fluxes whose measured values are classified asunreliable, this allows greater changes in the values than is the casewith presumably more accurately measured fluxes.

Before illustrating the application of MFA in bioprocess engineeringissues, the statistical validation of metabolic metabolic models, ascarried out according to embodiments of the invention, will first bedealt with in the following section, as this builds on theconsiderations just explained.

Validation of the Biochemical Metabolic Model

The quality of the postulated biochemical network model has not yet beenaddressed in the previous remarks. However, it is obvious that aninsufficient quality in its formulation may lead to severe deficits inthe results of MFA. Validation methods are needed to generate ameaningful model as a compromise between high significance and thegreatest possible simplification. In a publication by van der Heijden etal. from 1994, statistically motivated tests are presented and thedetection of possible systematic sources of error is explained (seebibliography [40] of the appendix). The investigations are based on theanalysis of flows that can be balanced and their influence oninconsistencies in the model. They are therefore only applicable if aredundant system is present.

A Test for the Evaluation of Inconsistencies

In the previous section in equation (2.11) the reduced form R′ of theredundancy matrix was already introduced. The residual vector is definedby

ε:=R′v _(m).

For a redundant system, the following generally applies ε≠0. Thecovariance matrix C_(ε) of ε may be calculated by

C _(ε) :=R′C _(v) _(m) ⁻¹ R′ ^(T)

It is therefore dependent on the covariance matrix of the measuredfluxes, which takes into account the uncertainties in the measurements.For the test, the test statistics H_(ε) are used, whose observations aregiven by:

h _(ε):=ε^(T) C _(ε) ⁻¹ε.

It may be shown that the test statistics are subject to aχ²-distribution (see [40] in the Appendix). The degrees of freedomcorrespond to the rank of C_(ε).

Overall, the following hypothesis test is obtained:

Test

H₀: The inconsistency of the considered metabolic model is notsignificantagainstH₁: The inconsistency of the considered metabolic model is significant:

Reject H ₀ at significance level Lehne H ₀ ab auf dem Signifikanzniveauα⇔h _(ε)>χ_(Rang(C) _(ε) _(),1−α) ².  (2.12)

χ_(Rang(C) _(ε) _(),1−α) ² denotes the (1−α)·100%-quantile of theχ²-distribution with Rang(C_(ε)) degrees of freedom.

Detection of Possible, Systematic Sources of Error

If the previously defined test indicates inconsistencies, this may bedue to an underestimation of the measurement noise, which is reflectedin the matrix C _(v) _(m) and thus influences the outcome of the test.Three further possible sources of error are discussed in the literature[40] of the appendix:

Systematic measurement errors: The measurement of the j-th flux v _(m,j)is subject to a systematic error π=v _(m,j)−v_(m,j).

Absence of an important reaction in the metabolic network: An (k+1)-thimportant reaction is missing in the network model; the stoichiometricmatrix A would have to be extended by another column a_(k+1). Thecorresponding flux is also indicated as v_(k+1).

Incorrect definition of a reaction in the metabolic network: The j-threaction is incorrectly defined; the vector a_(j)+Δa_(j) should be usedinstead of the column vector a_(j) in the stoichiometric matrix.

The investigation of the error is based on the structure of the residualvector ε: for each of the above errors, a characteristic comparisonvector υ may be defined, whose direction is approximately the same asthe direction of ∈, provided that the source of the error is actuallypresent. A statistical test which evaluates the similarity between thedirections of the vectors is also presented. The length of ε gives anindication of the size of the error s. In the following table thecorresponding comparison vectors are listed. r′_(j) denotes here thej-th column vector of R′. The derivation of the comparison vectors shallbe demonstrated here only for the first of the listed error sources. Forthe further cases please refer to [40].

The following applies

r′ ₁ v _(m,1) +r′ ₂ v _(m,2) + . . . +r′ _(k) _(m) v _(m,k) _(m) =ε.

n case of a correctly defined network model, the following applies

[ε]=0. If the true value for the j-th flux differs from the measured oneby the systematic error π, i.e.

v _(m,j) =v _(m,j)−π,

then

${\lbrack ɛ\rbrack} = {{E\left\lbrack {{r_{1}^{\prime}{\overset{¯}{v}}_{m,1}} + {r_{2}^{\prime}{\overset{¯}{v}}_{m,2}} + \ldots + {r_{j}^{\prime}\left( {v_{m,j} + \pi} \right)} + \ldots + {r_{k_{m}}^{\prime}{\overset{\_}{v}}_{m,k_{m}}}} \right\rbrack} = {{{\left\lbrack {{r_{1}^{\prime}{\overset{¯}{v}}_{m,1}} + {r_{2}^{\prime}{\overset{¯}{v}}_{m,2}} + \ldots + {r_{j}^{\prime}v_{m,j}} + \ldots + {r_{k_{m}}^{\prime}{\overset{¯}{v}}_{m,k_{m}}}} \right\rbrack} + {\left\lbrack {r_{j}^{\prime}\pi} \right\rbrack}} = {r_{j}^{\prime}{\pi.}}}}$

It is therefore to be expected that ε and r′_(j) have the samedirections.

TABLE 1 Comparison vectors and associated error sizes for threedifferent error sources. Comparison Error Error source vector r υ size sMeasurement of the j-th flux υ _(m,j) is false r′_(j) π One (k + 1)-threaction is missing Γ(A_(u)A_(u) ^(#)− 1)a_(k+1) υ_(k+1) The j-threaction is defined incorrectly Γ(A_(u)A_(u) ^(#)− 1)Δa_(j) υ_(j)

To assess the similarity between ε and υ, the test statistics

$\Delta^{2} = {{ɛ^{T}C_{ɛ}^{- 1}ɛ} - \frac{\left( {ɛ^{T}C_{ɛ}^{- 1}v} \right)^{2}}{v^{T}C_{ɛ}^{- 1}v}}$

are used, which are χ²-distributed with a degree of freedom ofRang(C_(ε))−1.

The following hypothesis test assesses the similarity of ε and υ:

Test

H₀: The vectors ε and υ are similaragainstH₁: The vectors are not similar:

Reject H ₀ at significance level Lehne H ₀ ab auf dem Signifikanzniveauα⇔Δ²>χ_(Rang(C) _(ε) _()−1,1−α) ².  (2.13)

The statistical derivation can be found in the appendix of reference[40].

According to embodiments, the metabolic model generated according toembodiments of the invention comprises a network, which should comprisethe central intracellular material fluxes and yet have a complexity aslow as possible. The model explained here as an example is essentiallybased on the network stoichiometries proposed in the followingpublications: Altamirano C, Illanes A, Becerra S, Cairo J J, Godia F(2006): “Considerations on the lactate consumption by CHO cells in thepresence of galactose”, Journal of Biotechnology 125, 547-556; LlanerasF, Pico J (2007): “A procedure for the estimation over time of metabolicfluxes in scenarios where measurements are uncertain and/orinsufficient”, BMC Bioinformatics 8:421; and Nolan R P, Lee K (2011):“Dynamic model of CHO cell metabolism”, Metabolic Engineering 13,108-124.

Compartments of the cells were not considered, however. Due to theirlarge number, not all reactions involving redox and energy equivalentsmay be included in the metabolic model. Therefore, NAD(P)H and ATP werenot included in the formulation of the stoichiometry. In addition, somemetabolic branches were not considered in detail but were integratedinto the biomass formation (e.g. the pentose phosphate pathway). In mostcases, the formulated reactions are a summary of several successivebiochemical reactions without branches, which should have identicalmaterial fluxes according to the steady-state assumption (for example,only a few intermediates of glycolysis or the citrate cycle areexplicitly listed).

The biomass balance was taken from the above mentioned publication byNolan (2011), as well as the conversion of the live and total celldensity into the unit mol/l. The formulation of stoichiometry forproduct formation follows from the amino acid composition of the targetprotein. The above mentioned publications were also used to determinethe reversibility of the reactions.

The resulting metabolic model is shown in detail in FIG. 4. This showsin FIG. 4a a biochemical network model of intra- and extracellularmaterial fluxes of CHO cells] {Biochemical network model of intra- andextracellular material fluxes of CHO cells It models the central fluxesof metabolism, which are the transport and conversion of glucose (Glc),lactate (Lac), alanine (Ala), glutamate (Glu), glutamine (Gln), ammonia(NH3), aspartate (Asp), comprise asparagine (Asn), serine (Ser), glycine(Gly), total cell density (BIO), product (Prod), glucose-6-phosphate(G6P), pyruvate (Pyr), alpha-ketoglutarate (AKG), malate (Mal) andoxaloacetate (Oxa) Reversibility is indicated by the shape of thearrows.

The table in FIG. 4b lists the individual stoichiometric reactions ofthe metabolic model. Of these, reactions 1, 3, 9, 11, 13, 14, 16, 18,20, 21, 22 and 23 are extracellular. The model comprises 13intracellular metabolites for which the steady-state assumption is toapply. The index “e” here refers to extracellular substances. Thereversibility/reversibility of reactions is indicated by the reactionarrow.

The network model contained in the metabolic model should comprise thecentral intracellular material fluxes and yet be as simple as possible.The formulation chosen in this thesis is essentially based on thenetwork stoichiometries proposed in the above mentioned publications byAltamirano et al (2006), Llaneras et al (2007) and Nolan et al (2011).Compartments of the cells were not considered, however. Due to theirlarge number, not all reactions involving redox and energy equivalentsmay be included in the metabolic model. Therefore, NAD(P)H and ATP werenot included in the formulation of the stoichiometry. In addition, somemetabolic branches were not considered in detail but were integratedinto the biomass formation (e.g. the pentose phosphate pathway). In mostcases, the formulated reactions are a summary of several successivebiochemical reactions without branches, which should have identicalmaterial fluxes according to the steady-state assumption (for example,only a few intermediates of glycolysis or the citrate cycle areexplicitly listed).

The biomass balance was taken from the above mentioned publication byNolan (2011), as well as the conversion of the live and total celldensity into the unit mol/I. The formulation of stoichiometry forproduct formation follows from the amino acid composition of the targetprotein.

The above-mentioned publications were also consulted regarding thereversibility of the reactions.

The stoichiometric matrix A was formulated for the metabolic networkshown in the table above. The columns corresponding to the known (inthis case extracellular) material fluxes were combined to form thesubmatrix A_(m), the others to the submatrix A_(u).

The characterization of the metabolic network may be carried outaccording to a scheme which is shown and explained in the appendix asFIG. 4.2.

The metabolic network may be validated and, if necessary, modified asdescribed in the appendix.

Thus, a metabolic model 402 of CHO cells has been provided as shown inFIG. 4. The metabolic model includes a variety of intracellular 410 andextracellular 408 fluxes, and the metabolic model specifies at least onestoichiometric relationship between an intracellular 406 and anextracellular 404 metabolite.

The following steps 106-112 are performed for a plurality of points intime during the cultivation of a cell culture in a bioreactor. A profileof actually measured extracellular material fluxes and extracellularfluxes predicted for the next point in time (after an interval ofdefined length, e.g. 24 h) may be generated. The deviation of these twoprofiles from each other indicates the quality of the prediction.

Step 106: Receiving Measured Values

In one embodiment, a sample is taken at several points in time duringthe cultivation of a cell culture in a bioreactor 208 of that cellculture and transferred automatically or manually to one or moreanalysing devices 250 as shown in FIG. 2. The analysing device may be asystem of one or more analysers, for example a Thomas chamber or anoptical counting station for determining cell density. For example, ahigh performance liquid chromatograph or other suitable methods known inthe state of the art may be used to determine the concentration ofindividual amino acids. Samples may be taken, for example, at 24-hourintervals. The measured data obtained in this way are transmitted to adata processing system 252. The data processing system 252 may, forexample, be a computer which, as a control unit, monitors and/orcontrols one or more bioreactors.

According to some embodiments, at least some of the measured values, forexample the cell density, are also determined by corresponding sensorsof the bioreactor 208 itself and transmitted to a data processing system252.

In addition to the concentrations of selected extracellular metaboliteswhich are known or expected to have a certain predictive power withrespect to the concentration and flux of this or another extracellularmetabolite at a future point in time, other input parameter values mayalso be determined, in particular the current time, the current celldensity, and, if appropriate, other parameters such as the LDHconcentration, which may be used as a correction factor for the lysedcells not included in the cell density determination. The measured datathus obtained empirically at a determined point in time may now be usedfor the predictions of extracellular fluxes at a subsequent point intime, for example the next day, by means of an MLP, as described in thefollowing step.

Step 108: Input of the Measured Values into a Trained MLP

The data processing system 252 includes an MLP, for example a neuralnetwork (NN) or a cooperating system of several neural networks, whichhas been trained to predict or estimate one or more extracellular fluxesof the metabolic model 254 on the basis of input parameter values (inparticular concentrations of extracellular metabolites and cell density)measured at a determined point in time. For example, the data processingsystem 252 may include a program logic which automatically transfers themeasured data obtained at a point in time as input to an MLP trained ontest data sets obtained from cell cultures of the same type of cells asthe cells of the cell culture whose metabolic state is to be predictedat a future point in time (for example, next day).

Step 110: Predictions of the MLP's Future Intake and Release Rates

Using the neural network, extracellular fluxes are predicted orestimated in a one-step-prediction based on currently measuredconcentrations of extracellular metabolites c _(m,j) ^((n)) at the pointin time t^((n)) at a future point in time t^((n+1)) chosen at will.Thus, in this step, in response to the input of the measured inputparameter values into the MLP, the MLP calculates and returns one ormore extracellular fluxes for the future point in time.

Optional: Predictions of the MLP of Concentrations of ExtracellularMetabolites

The extracellular fluxes estimated via the neural network are, accordingto embodiments of the invention, also used for one-step predictions ofmetabolite concentrations at an arbitrarily chosen future point in timet^((n+1)) based on the current concentrations c _(m,j) ^((n)) at thatpoint in time t^((n)).

For this purpose, equation (4.1) of the appendix is solved according toc2. Since the live cell density at the future point in time VCD^((n+1))is unknown, it is replaced by the currently measured and, if necessary,corrected cell density. The reformulated equation is:

$\begin{matrix}{{\overset{\Cup}{c}}_{m,j}^{({n + 1})} = {{{{\overset{\Cup}{v}}_{m,j}^{(n)}\left( {t^{({n + 1})} - t^{(n)}} \right)}\frac{{VCD}^{(n)}\left( {V^{(n)} + V^{({n + 1})}} \right)}{2V^{({n + 1})}}} + {{\frac{{{\overset{\_}{c}}_{m,j}^{(n)}V^{(n)}} + {\text{?}V\text{?}}}{V^{({n + 1})}}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (4.4)\end{matrix}$

If feedings are carried out during the operation of the bioreactor, theyshould preferably be considered in c_(zu) _(m,j) ^((n)), V_(zu) ^((n))and V^((n+1)).

Step 112: Implementation of a MFA

Based on the uptake and release rates of the extracellular metabolites(extracellular fluxes) as predicted by the MLP for the future point intime, a metabolic flux analysis is then carried out in accordance withthe embodiments of the invention, which also incorporates theintracellular fluxes and stoichiometric equations as formulated in themetabolic model. Since in the metabolic model extracellular andintracellular fluxes are coupled to each other via one or moreintracellular metabolites, it is mathematically possible to also predictthe intracellular fluxes at the future point in time on the basis of thepredicted extracellular fluxes. Corresponding program routines forperforming metabolic flux analysis can be implemented in Matlab andother software solutions available on the market.

Since the predictions of the intracellular fluxes include both thepredictions of the MLP trained on dynamic, empirical data and theknowledge of stoichiometric relationships and reaction equationsspecified in the metabolic model, this prediction step may also bedescribed as a prediction of a hybrid model relationship.

For example, the coupling of the results of the predictions of the MLPwith the information of the metabolic model in the course of materialflow analysis may be implemented as follows

After the prediction of the extracellular fluxes in the following timeinterval using the neural network, a metabolic material flow analysis isperformed to estimate the flux distribution in the next time interval.

First, an estimate of a covariance matrix of the fluxes of the metabolicmodel is generated:

C _({hacek over (v)}) _(m) =

[(v _(m) −{hacek over (v)} _(m))(v _(m) −{hacek over (v)} _(m))^(T)]

The formulation

${\left\lbrack {\left( {v_{m} - \overset{\Cup}{v}} \right)\left( {v_{m} - {\overset{\Cup}{v}}_{m}} \right)^{\top}} \right\rbrack} = {{\left\lbrack {\left( {v_{m} - \;  + {\overset{\_}{v}}_{m} -} \right)\left( {v_{m} - + \text{?} - {\overset{\Cup}{v}}_{m}} \right)^{\top}} \right\rbrack} = {\underset{= \text{?}}{\underset{}{\left\lbrack {\left( {v_{m} - \text{?}} \right)\left( {v_{m} - \text{?}} \right)^{\top}} \right\rbrack}} - {2{\left\lbrack {\left( {v_{m} - \text{?}} \right)\left( {{\overset{\Cup}{v}}_{m} - \text{?}} \right)^{\top}} \right\rbrack}} + {\left\lbrack {\left( {\text{?} - \text{?}} \right)( - )^{\top}} \right\rbrack}}}$?indicates text missing or illegible when filed

shows, that both the quality of the measurements and the quality of theestimates of the measured values by the neural network are incorporatedinto the covariance matrix. However, the rewording may not facilitatethe estimation, since the differences (v_(m)−v _(m)) and ({hacek over(v)}_(m)−v _(m)) do not represent independent random variables and theexpected value in the middle term of the right side and thus theexpected value in the middle term of the right side is unknown.Therefore, the expression used in this paper according to Equation 4.5in the Appendix, which will be explained in the following, is only arough approximation of the actual covariance matrix.

According to embodiments, the covariance matrix is chosen as a diagonalmatrix for the purpose of predicting future intracellular fluxes, sincethe different metabolite fluxes are estimated over separate networks andthe errors may therefore be considered largely independent of eachother. By definition, the diagonal entries should reflect, or at leastbe proportional to, the variations in the errors of the estimated fluxes(the proportionality factor does not play a role in solving the MFAproblem as shown in Equation 2.9 of the Appendix).

In contrast, according to embodiments of the invention for descriptivemetabolic material flow analysis of the current metabolic state of acell, a covariance matrix C_(v) _(m)_() is used in which the diagonal entries are not dependent on the amount of fluxes.)

Be {hacek over (υ)}_(m,j) ^((n)) the j-th measured predictive flux inthe n-th time interval. The covariance matrix was formulated as adiagonal matrix and has the structure:

$\begin{matrix}{C_{{\overset{˘}{v}}_{m}} = {\begin{pmatrix}\psi_{1} & \; & 0 \\\; & \ddots & \; \\0 & \; & \psi_{k_{m}}\end{pmatrix}.}} & (4.5)\end{matrix}$

The diagonal entries ψ_(j) were chosen as the medians of the quantities

{(υ _(m,j) ^((n))−{hacek over (υ)}_(m,j) ^((n)))² |n Zeitintervall inTrainingsgatensatz}  (4.6)

In this embodiment it is assumed that the “time interval in the trainingdata set” is identical or similar to the time interval to be used forthe current prediction.

A flow analysis is then performed on the basis of this covariance matrixas is known per se in the state of the art. The covariance matrix isused to balance the fluxes according to equation (2.10) of the appendix.Equation 2.10 of the Appendix refers to the descriptive MFA, where adifferent covariance matrix was used according to the embodimentdescribed in the Appendix. However, the balancing or calculation of thefluxes by MFA is done in the same way in the case of predictive MFA.

Optionally, an error analysis of the model based e.g. on a Gaussianerror propagation may be performed as described in section 4.7.8 of theAppendix.

Generation and Training of a Neural Network

Compared to the specification of reaction-kinetic models for theprediction of future material fluxes, the use of trained MLPs has theadvantage that their generation is usually easier and faster in asemi-automatic method. An example of how an MLP in the form of an NN maybe generated by training is described below.

a) Cultivation of Several Training Cell Cultures

In order to obtain the broadest possible database for MLP training,several training cultures are preferably cultivated in severalbioreactors. Preferably, these bioreactors comprise one or morefed-batch reactors and one or more additional bioreactors from otherreactor types.

Eight fermentations of a clone of recombinant CHO-cells in an embodimenthave been carried out on a one-litre scale. The initial conditions(volume, media composition, inoculum concentration) were chosenidentically for each bioreactor, but different operation modes wereused:

-   -   A bioreactor was operated in a batch mode until the viability of        the cells dropped below 50%.    -   In a second bioreactor, the batch method was also used        initially. Towards the end of the exponential growth phase, a        partial harvest was carried out and the reactor was filled up        with fresh medium, so that conditions similar to those at the        beginning of the fermentation were achieved (in terms of volume        and inoculum concentration). Subsequently, the batch method was        continued (so-called split-batch method).    -   The fermentation in the remaining six reactors was carried out        as a fed-batch with an initial batch phase. Both continuous        feeding and a pulse-like nutrient addition took place towards        the end of the fermentation. The fed-batch fermentations        differed in the glucose concentrations in the medium, which were        adjusted by different feeding strategies. In the following        explanations, the fed-batch approaches will often be numbered.        According to this numbering, in the first two approaches a short        term complete glucose limitation took place in the second half        of the process before the bolus additions were made. The third        and fourth approaches had the same limitation, but the        subsequent boluses set higher glucose concentrations. In the        fifth and sixth approaches there was always a positive minimum        glucose concentration.

Temperature, pO2 value and pH value were kept constant during the entirefermentation. Regular sampling was carried out throughout the process.The samples were examined with regard to their live and total celldensity, using a staining method that distinguishes between living anddead cells. Lysed cells were not recorded. In addition, the content ofvarious substances in the reaction medium was examined using a COBASINTEGRA analyzing device or high-performance liquid chromatography.These included glucose, lactate and ammonium as well as the amino acidsalanine, glutamine, glutamate, asparagine, aspartate, serine, glycineand the enzyme lactate dehydrogenase. The product concentration was alsodetermined.

At the sampling points, the volume of liquid in the bioreactor wasmeasured using a fermenter scale.

b) Determination of the Network Architecture

A neural network was generated which was to estimate the mean fluxes ofextracellular metabolites between the current and the next sampling timefrom the current state in the bioreactor (so-called one-stepprediction). According to some embodiments, a separate network wastrained for each extracellular material flux as an output variable. Aselection of the currently prevailing extracellular metaboliteconcentrations served as input variables.

The neural network consisted of a two-layer perceptron with linearactivation function in the output layer and sigmoidal activationfunction in the hidden layer (see FIG. 2.4 in the appendix). For thelatter, both the sigmoid function and the hyperbolic tangent weretested. There were no significant differences to the results obtainedusing the sigmoid function. The training was performed according to thesequential gradient descent method (as exemplified in equations (2.16),(2.17) and (2.18) in the appendix).

c) Selection of Input Variables

According to preferred embodiments of the invention, several or allextracellular metabolites mentioned in the metabolic model are sortedaccording to their relevance for the prediction of the respective fluxin order to select the input parameter values. Preferably, extracellularmetabolites with redundant information content were not considered.

A detailed description for the selection of the input parameter valuesaccording to the embodiments of the invention is given in thedescription of FIG. 13.

d) Selection of Training Parameters, Initialization of Weights andNormalization of Data

After the number of iterations and hidden neurons has been determined,the values η=0, 1 in the output layer and η=0,02 in the hidden layer arenow selected for the initial learning step in each net. After each tenthof the total number of iterations, the step sizes are reduced by 1/10 ofthe initial value. Initial values for the weights were generated to [−1/10, 1/10] equally distributed random numbers. The input and outputtraining data were standardized separately by metabolite so that theadjusted values had the empirical mean 0 and the empirical standarddeviation 0.5. The test data were transformed in the same way with themean values and standard deviations of the training data set.

e) Selection of Training and Test Data Sets

The estimates were made by three different neural networks, which differin the grouping of the data into training and test data sets:

-   -   1. network 1: The training data set of the first network        consisted of the data from three of the fed-batch fermentations        and the batch fermentation, the test data set consisted of the        data from the other three fed-batch fermentations and the        split-batch fermentation.    -   2. network 2: The second network had data from three of the        fed-batch fermentations in the training data set and the data        from the remaining three of the fed-batch fermentations, batch        fermentation and split-batch fermentation in the test data set.    -   3. network 3: The training data set of the third network        comprised the data from three of the fed-batch fermentations and        the test data set comprised the data from the remaining three        fed-batch fermentations.

f) One-Step Prediction of Extracellular Metabolite Concentrations

The goal is the generation of a trained MLP, which allows the mostaccurate predictions of the metabolism of a cell.

If low-frequency data is generated during the generation of the trainingdata set, it may happen that averaging/filtering of the data would leadto a too large loss of information. Therefore, in this case, meanextracellular fluxes between two consecutive measurement points shouldbe approximated. The calculation is based on equation

{dot over (m)}={dot over (V)} _(zu) ·c _(zu) −{dot over (V)} _(ab) ·c_(ab) +Q

and will be explained in the following:

Since the batch and the fed-batch method are to be considered and thereis therefore no liquid discharge, the following applies in any case {dotover (V)}_(ab)=0. The extracellular flux v of a component at time t is,as already mentioned in section 2.2.2 of the Appendix, the amount ofsubstance that is absorbed or released by a cell per time. It istherefore given by

${\upsilon (t)}:={\frac{Q(t)}{{VC}(t)}.}$

where VC(t) is the number of living cells in the reaction medium at timet. According to equation 2.1 in the appendix, the following thereforeapplies in the area between two discontinuities

${\upsilon (t)} = {\frac{{\overset{.}{m}(t)} - {{{\overset{.}{V}}_{zu}(t)} \cdot c_{zu}}}{{VC}(t)}.}$

The concentration of the substance in the feed is constant over time oris assumed to be approximately constant.

Unsteadiness may occur at the sampling times, and during bolusadditions.

Initially it should be assumed that the addition of nutrients is alwayscontinuous and that therefore the quantities m and V_(zu) can bedifferentiated between two measuring points. If one wants to determinethe mean flux υ_(avg) between two consecutive measuring points t1 andt2, one may estimate it—at first seemingly trivial—by:

$\begin{matrix}{\upsilon_{avg} = {\frac{\frac{\Delta \; m}{\Delta \; t} - {\frac{\Delta \; V}{\Delta \; t}c_{zu}}}{{VC}_{avg}} = {\frac{m_{2} - m_{1} - m_{zu}}{\left( {t_{2} - t_{1}} \right)\frac{{VC}_{1} + {VC}_{2}}{2}} = \frac{{c_{2}V_{2}} - {c_{1}V_{1}} - {c_{zu}V_{zu}}}{\left( {t_{2} - t_{1}} \right)\frac{{{VCD}_{1}V_{1}} + {{VCD}_{2}V_{2}}}{2}}}}} & (4.1)\end{matrix}$

Here, the indexed variables denote the value at point in time t1 or t2,m_(zu) is the amount of substance that is fed into the reaction mediumvia the feed in the considered time interval and VCD denotes the livingcell density. The operator Δ symbolizes the difference of thecorresponding quantities between t1 and t2.

The above estimation (4.1) shall be mathematically substantiated in thefollowing. It is obtained by integration over the time interval (t₁,t₂), whereby the individual measured quantities are linearlyinterpolated between the two sampling times, and by additional Taylordevelopment:

With linear interpolation, the following applies to all t∈(t₁, t₂):

${\overset{.}{m}(t)} = \frac{m_{2} - m_{1}}{t_{2} - t_{1}}$${\overset{.}{V}}_{zu} = \frac{V_{2} - V_{1}}{t_{2} - t_{1}}$${{VC}(t)} = {{{VC}_{1} + {\left( {{VC}_{2} - {VC}_{1}} \right)\frac{t - t_{1}}{t_{2} - t_{1}}}} = {\frac{{\left( {t_{2} - t} \right){VC}_{1}} + {\left( {t - t_{1}} \right){VC}_{2}}}{t_{2} - t_{1}}.}}$

Therefore

$\begin{matrix}{\upsilon_{avg} = {\frac{1}{t_{2} - t_{1}}{\int_{t_{1}}^{t_{2}}{\frac{Q(t)}{{VC}(t)}{dt}}}}} \\{= {\frac{1}{t_{2} - t_{1}}{\int_{t_{1}}^{t_{2}}{\frac{{\overset{.}{m}(t)} - {{(t) \cdot c_{zu}}}}{{VC}(t)}{dt}}}}} \\{= {\frac{1}{t_{2} - t_{1}}{\int_{t_{1}}^{t_{2}}{\frac{\frac{m_{2} - m_{1}}{t_{2} - t_{1}} - {c_{zu}\frac{V_{2} - V_{1}}{t_{2} - t_{1}}}}{{VC}(t)}{dt}}}}} \\{= {{\frac{m_{2} - m_{1} - {c_{zu}\left( {V_{2} - V_{1}} \right)}}{t_{2} - t_{1}} \cdot \frac{1}{t_{2} - t_{1}}}{\int_{t_{1}}^{t_{2}}{\frac{1}{{VC}(t)}{dt}}}}} \\{= {{\frac{m_{2} - m_{1} - m_{zu}}{t_{2} - t_{1}} \cdot \frac{1}{t_{2} - t_{1}}}{\int_{t_{1}}^{t_{2}}{\frac{1}{{VC}(t)}{dt}}}}}\end{matrix}$

The following applies to the remaining integral:

$\begin{matrix}{{\frac{1}{t_{2} - t_{1}}{\int_{t_{1}}^{t_{2}}{\frac{1}{{VC}(t)}{dt}}}} = {\int_{t_{1}}^{t_{2}}{\frac{1}{{\left( {t_{2} - t} \right){VC}_{1}} + {\left( {t - t_{1}} \right){VC}_{2}}}{dt}}}} \\{= {\int_{t_{1}}^{t_{2}}{\frac{1}{{t_{2}{VC}_{1}} - {t_{1}{VC}_{2}} + {t\left( {{VC}_{2} - {VC}_{1}} \right)}}{dt}}}} \\{= {\frac{1}{{VC}_{2} - {VC}_{1}}\left\lbrack {\ln {{{t_{2}{VC}_{1}} - {t_{1}{VC}_{2}} + {t\left( {{VC}_{2} - {VC}_{1}} \right)}}}} \right\rbrack}_{t_{1}}^{t_{2}}} \\{= \frac{{\ln \left( {\left( {t_{2} - t_{1}} \right){VC}_{2}} \right)} - {\ln \left( {\left( {t_{2} - t_{1}} \right){VC}_{1}} \right)}}{{VC}_{2} - {VC}_{1}}} \\{= \frac{{\ln \left( {VC}_{2} \right)} - {\ln \left( {VC}_{1} \right)}}{{VC}_{2} - {VC}_{1}}}\end{matrix}$

A Taylor expansion of the logarithms by

$\frac{{VC}_{1} + {VC}_{2}}{2}$

up to the first order is performed, which results in:

$\begin{matrix}{\frac{{\ln \left( {VC}_{2} \right)} - {\ln \left( {VC}_{1} \right)}}{{VC}_{2} - {VC}_{1}} = \frac{{\ln \left( {\frac{{VC}_{1} + {VC}_{2}}{2} + \frac{{VC}_{2} - {VC}_{1}}{2}} \right)} - {\ln \left( {\frac{{VC}_{1} + {VC}_{2}}{2} + \frac{{VC}_{1} - {VC}_{2}}{2}} \right)}}{{VC}_{1} - {VC}_{2}}} \\{\approx \frac{{\ln \left( \frac{{VC}_{1} + {VC}_{2}}{2} \right)} + {\frac{{VC}_{2} - {VC}_{1}}{2}\frac{2}{{VC}_{1} + {VC}_{2}}} - {\ln \left( \frac{{VC}_{1} + {VC}_{2}}{2} \right)} - {\frac{{VC}_{1} - {VC}_{2}}{2}\frac{2}{{VC}_{1} + {VC}_{2}}}}{{VC}_{1} - {VC}_{2}}} \\{= \frac{\frac{{2\; {VC}_{2}} - {2\; {VC}_{1}}}{{VC}_{1} + {VC}_{2}}}{{VC}_{1} - {VC}_{2}}} \\{= \frac{1}{\frac{{VC}_{1} + {VC}_{2}}{2}}}\end{matrix}$

Using this expression for the above integral gives the estimate of theflux according to equation 4.1 above or the appendix).

If, between two sampling points in time t1 and t2, a bolus containingthe substance under consideration is added at time tB, the mean fluxbetween t1 and tB and between tB and t2 is assumed to be the same. Itmay then easily be shown that equation (4.1) may also be applied in thiscase, with the amount of substance added via the bolus being included inthe expression m_(zu).

Thus, at the point in time of a current sampling from a trainingbioreactor, both the concentrations of extracellular metabolites in theprevious sampling and the calculated extracellular fluxes calculated onthe basis of the extracellular fluxes calculated since the last samplingare known and may be transferred together as reference value quantitiesto the MLP to be trained, which thereby learns, on the basis of theconcentrations of extracellular metabolites measured in the lastsampling, to predict the calculated extracellular ones in such a waythat there is the smallest possible deviation from the calculatedextracellular fluxes.

The trained MLP or the trained neural network may now be stored and usedfor one-step predictions of extracellular metabolite concentrations atany chosen future point in time t^((n+1)), for example the next day,based on the current concentrations c _(m,j) ^((n)) at that point intime t^((n)).

In summary, the idea of training the neuronal network is based on thefact that the concentrations of extracellular metabolites are easilymeasurable and from these, at least in retrospect, extracellular fluxescan be determined empirically. By using the concentrations ofextracellular metabolites measured at a determined point in time asinput parameter values and the extracellular fluxes, as they can becalculated over the time interval between this current point in time anda future point in time based on the concentration difference of anextracellular metabolite, as output parameter values, a neural networkor in other machine learning algorithms may be trained to predict atleast the extracellular fluxes for the one future point in time. Thedetermination of the cell density allows an allocation of the totalconcentration difference in the medium to the individual cells of thecell culture contained in the medium.

The uptake/discharge rate of extracellular metabolite (reaction term) iscalculated according to preferred embodiments from the difference of thetotal change of concentration of the substance in the bioreactor minusthe substance added to/removed from the bioreactor (convection term). Itwas found that the reaction term for extracellular metabolites isdetermined primarily from the uptake or release into or through cells,so that the measured concentration changes of the extracellularmetabolites may be essentially equated with the uptake or release ratesof these extracellular metabolites into or from the cells. However, insome embodiments which provide for a significant supply or removal ofcertain extracellular metabolites during operation of the bioreactor, acorrective calculation may be made by subtracting from the measuredconcentration changes those parts of the concentration changes which aredue to the external supply or removal of the extracellular metabolitesto or from the culture medium when calculating the uptake or removalrates of these extracellular metabolites. However, this correction doesnot necessarily have to be made even in the Fed-Batch method. Also inthe Fed-Batch method the incoming and outgoing flows are continuous. Inthe case of bolus feeding or sampling during fermentation, one obtainscurves which are continuous over long distances and thus differentiable.This justifies the assumption of a differentiable course of the changesin concentration also for fed-batch reactors.

However, it is preferable to correct the measured cell density duringtraining and/or in the predictions of the extracellular fluxes using thetrained MLP. With the exception of batch fermentation, there seems to bea similar, approximately linear relationship between LDH concentrationand cell density difference in fermentation approaches.

According to the design of the invention, a measured total cell densityis continuously recorded in a fermenter and presented in a plot. Inparallel, the cell density is calculated. The predictions may becalculated, for example, by a trained MLP trained according toembodiments of the invention, using the measured cell concentration as afurther output parameter value. This “predicted” cell density is alsoplotted in the plot. Thus, according to embodiments of the invention, afirst temporal profile of the measured cell density of a cell culture ofa certain cell type is empirically determined and also a second temporalprofile of a predicted cell density based on the metabolic model and theextracellular metabolite concentrations. Thus, a plot is obtained whichcontains a temporal profile of the measured and predicted cell densitiesand their deviation from each other.

It has been shown that there is often a considerable deviation betweenmeasured cell density and the cell density predicted by MLP, especiallytowards the end of a cell culture. The discrepancy between measured andpredicted cell density is referred to in the following as the “densitydiscrepancy profile” and may optionally also be shown graphically in theplot. The density discrepancy profile represents the temporal profile ofthe occurrence of lysed cells, which are not measurable as cells butstill have an influence on the concentrations of extracellularmetabolites. An empirical function, e.g. with the help of inner linearcompensation lines over the density discrepancy profile characterized bytwo parameters and, is then generated, which sets the density of lysedcells according to the density discrepancy profile in linear relation tothe LDH concentration in the medium measured at a determined point intime. This function allows an approximate conversion from the LDHconcentration to the density difference and thus to the lysed cells. Thecorrected cell density is therefore the sum of the measured cell densityand the number of lysed cells calculated by the linear function based onthe measured LDH concentration. In other words, the measured celldensities c _(m,11) ^((n)) are supplemented by the density differenceusing this

c _(lys,11) ^((n)) =c _(m,11) ^((n)) +a ₁ c _(LDH) ^((n)) +a′ ₀,

where c _(LDH) ^((n)) the LDH concentration at the n-th point in time isdesignated and a′₀ chosen so that c _(lys,11) ^((n) ⁰ ⁾=c _(m,11) ^((n)⁰ ⁾ is applied if n₀ is a point in time when a fermentation was started.

These corrected cell density values are used to calculate the correctedbiomass fluxes v _(lys,11) ^((n)) according to preferred executionforms. This has the advantage that the proportion of inconsistent fluxdistributions as a result of MFA is significantly reduced.

FIG. 2 shows an example of the process of information acquisition inseveral stages using different devices and data sources.

Cells whose metabolic state is to be predicted at a future point in timeare kept in a bioreactor 208, for example in a fed-batch fermenter. Thefermenter may contain some sensors or be operationally coupled to them,for example sensors for determining cell density. Instead of or inaddition to these sensors, samples may be taken regularly from the cellculture and transferred to one or more analytical instruments. There theconcentration of extracellular metabolites is measured. The measuredcell density, the measured extracellular metabolite concentrations and,where appropriate, the point in time and amount of external supply ofmetabolites (e.g. glucose boli) are transferred to a data processingsystem 252. This system 252 comprises a metabolic model of the cell andan MLP trained to make predictions of the extracellular fluxes based oncurrently measured extracellular metabolite concentrations for thecurrently used cell type. The measured values received and measured at adetermined point in time are transmitted as input to the trained MLP,which then predicts extracellular fluxes at a future point in time. Inthe course of a subsequent MFA, the extracellular fluxes and thestoichiometric equations of the model are also used to predict theintracellular fluxes for the future point in time. The complete modelincluding the predicted extracellular and intracellular fluxes may bedisplayed and/or stored as a “snapshot” image 254 of the metabolic stateof a cell via a graphical user interface.

If the bioreactor 208 contains a training cell culture, i.e. a cellculture from which data are regularly collected over a longer period oftime in order to generate a training data set, the data processingsystem 252 is additionally adapted to calculate an extracellular fluxfor the future time interval (i.e. the time interval from the currentpoint in time to the next point in time for which a prediction of themetabolic state is to be made) on the basis of a plot of the measuredconcentrations of extracellular metabolites and to transfer this to theMLP as an output parameter value during training.

FIG. 3 shows a block diagram of a system for predicting the metabolicstate of a cell, which may be used to monitor and/or control one or morebioreactors.

The System 200 may be a data processing system of various types. Forexample, it may be a desktop computer, a server computer, a notebook ora user's portable mobile device. The System 200 may be a control modulethat is part of or connected to a bioreactor or bioreactor plant withmultiple bioreactors. In the embodiment shown here, the system iscoupled to three bioreactors 204, 206, 208 and is configured to monitorthe metabolic state of the cells in the respective cell culture in realtime and to predict for a future point in time, for example a point 12hours or 24 hours in the future. Each of the bioreactors may have one ormore measuring devices or sensors for determining cell density and/ormetabolite concentration or a mechanism that allows for sample naming sothat the metabolite concentration can be determined by other devices byanalysing the sample. Preferably, the 204-208 bioreactors have variouscontrol units such as valves, pumps, dosing units for boluses, stirrers,etc. which are coupled to the system and may receive and execute controlcommands from the system if necessary.

The System 200 includes one or more processors 202 as well as a firstinterface 210 for receiving measurement data from the one or morebioreactors. The interface 210 may be adapted as a direct interface tothe bioreactors or as an interface to analyzing devices in which samplesfrom the bioreactors are analyzed, or to a graphical user interface thatallows a user to enter the obtained measurement data manually or byother means.

In addition, the system comprises or is coupled to 201 volatile ornon-volatile storage medium 212. The storage medium may be, for example,main memory, hard disk, or memory of a cloud service, or networkstorage, or combinations of the above types of storage. The storagemedium contains a metabolic model 214 of the cells held and proliferatedin the bioreactors, for example a model as shown in FIG. 4.

The storage medium comprises a trained MLP 218, for example a trainedneural network, adapted to predict one or more extracellular fluxes at afuture point in time from the measured concentrations of severalextracellular metabolites received at a future point in time.

In addition, the storage medium comprises a program logic 220, which isadapted to transfer the received measured values to the MLP 218 in orderto perform a prediction of extracellular fluxes. Furthermore, theprogram logic is adapted to perform a real-time metabolic flux analysis(MFA) based on the metabolic model 214 and the predicted extracellularfluxes in order to predict intracellular fluxes for the future point intime. The program logic 220 may be implemented in any programminglanguage, e.g. C++, Java, Matlab, or in the form of several programmodules in different or the same programming language that areinteroperable with each other.

Optionally, the storage medium may contain several reference values andreference value ranges. These reference values or reference value rangesindicate acceptable or desirable intracellular fluxes of differentintracellular metabolites. By real-time comparison of the predictedintracellular fluxes with the reference values 216, program logic 220may detect whether the cells in one or more of the bioreactors areheading towards an undesirable metabolic state and, if necessary, takecountermeasures by issuing appropriate control commands to therespective reactor via a second interface 222 to counteract thepredicted trend. Alternatively or in addition, in this case a warningmay be given to a user via a user interface 224, for example a displaydevice, for example an LCD display. The display device may inform theuser of the predicted extracellular and intracellular fluxes and also ofany predicted deviations of these fluxes from desirable referenceranges.

FIG. 5 shows the calculation of intracellular fluxes at severalsuccessive points in time during the operation of a bioreactor. Theupper plot 502 shows a profile of the concentration of an extracellularmetabolite determined at six measurement points (one measurement perday). The middle plot 504 shows that a trained MLP uses thesemeasurements to predict one or more extracellular fluxes for a futurepoint in time. A comparison of the positions of the points in the upperand middle plot shows that the points in time at which the measurementdata were collected and the points in time at which the extracellularfluxes were predicted are about half a day apart.

This means that if, for example, the concentrations of extracellularmetabolites are measured daily at 12:00 noon, these data are used topredict the extracellular fluxes at midnight. The lower plot 506 showsthat by performing an MFA at each of these future points in time thepredicted extracellular fluxes were supplemented by predictedintracellular fluxes.

FIG. 6 shows different fluxes that are represented in the metabolicmodel shown in FIG. 4. The fluxes were calculated on the basis of themeasured change in metabolite concentrations over a time interval andthe measured cell density and show that the fluxes for differentmetabolites have a very different and partly characteristic course.

FIG. 7 illustrates the successful use of the method for generatingbiological knowledge. The plot of the glucose flux on the upper leftshows that in the profile of the glucose flux at about point in time0.65 the glucose flow comes to a virtual standstill. Effects of thisglucose deficiency can be observed for alanine, series and glycine:Alanine is absorbed at higher rates in the case of the limitations,which probably serves to provide more pyruvate. In addition, a strongerconversion from serine to glycine takes place, which is associated withincreased ammonium formation.

A comparison of the glucose and product flow curves reveals certainsimilarities: both curves show a temporary decrease of the fluxes at anearly point of fermentation as well as a later collapse when glucose wastemporarily absent from the medium. This dip is missing in the productflows of the fermenters without glucose limitation. Similarly, theeffect of the glucose boli, which shows the sensitivity of the glucoseflux to the glucose concentration in the medium, can also be seen in theproduct formation. Obviously there is a very close connection betweenthese fluxes. At the end of the fermentation process, the amount ofsubstance in the product was highest in those bioreactors withoutglucose limitation. Therefore, the availability of glucose seems to beessential for effective product formation, and a shortage should bestrictly avoided.

FIG. 8 illustrates another successful use of the method for generatingbiological knowledge.

The so-called lactate shift refers to the effect frequently observed incell culture cultivation that the lactate flow changes the sign frompositive to negative. There are numerous attempts in the literature toexplain the lactate shift. Mulukutla et al. postulate, for example, thatthe lactate shift is the result of regulatory mechanisms that are set inmotion by increasing lactate inhibition. This biological hypothesis wastested by determining the lactate flux in several bioreactors overseveral measurements using the method according to the invention and bymeasuring the extracellular lactate concentrations. The correspondingresults are shown in FIG. 9 for four bioreactors. It was shown that atthe point in time when a reversal of the net flow direction of thelactate was observed, the lactate concentration in the differentbioreactors was different. Thus, the mechanism postulated in theliterature up to that point does not seem to be responsible for thelactate shift. Rather, it seems as if the shift depends on the glutaminemetabolism, as shown in FIG. 6.2 of the appendix. A comparison of thecourses of glutamine concentrations and lactate flows in severaldifferent fermenter types shows that the lactate shift is alwaysaccompanied by a complete consumption of glutamine.

This observation has also been reported in the literature, where adetermined CHO clone was described in which the shift occurred onlyafter glucose was consumed (Zagari F, Jordan M, Stettler M, Broly H,Wurm F M (2013): “Lactate metabolism shift in CHO cell culture: the roleof mitochondrial oxidative activity”, New Biotechnology, Vol. 30, No.2). According to the invention, it is thus possible, by repeatedlypredicting intracellular fluxes and by comparing these fluxes with otherintracellular fluxes and/or concentrations of extracellular metabolites,to successfully test and, if necessary, reject hypotheses regarding cellmetabolism and to identify metabolic peculiarities of individual cellclones. Based on the results of the analysis, it is obvious that themajority of the lactate formed in the common CHO clones originates fromglutaminolysis. Interestingly, after the lactate shift, glutaminolysisinitially comes to a standstill: no more glutamine uptake takes place,rather it is formed in small quantities, which increases itsconcentration in the bioreactor. The cells now seem to have adjustedtheir metabolism exclusively to the substrate glucose. It is noticeablethat a second phase of glutamine uptake may be observed as soon asglucose reaches very low concentrations. At the same time, a short phaseof renewed lactate production can also be observed.

FIG. 9 shows plots with lactate fluxes and glutamine concentrations offour bioreactors of different types, which allows a comparison of thecourses of glutamine concentration and lactate flow in a determined cellculture. The glutamine concentrations are indicated by solid lines andthe lactate fluxes by dotted lines. The lactate fluxes andconcentrations of the individual fluxes and metabolites were previouslynormalized to increase comparability. The arrows indicate a reversal ofthe sign of the lactate flux.

FIG. 10 shows the time courses of intracellular fluxes as calculated bydescriptive modelling using MFA based on the metabolic model for thecurrent point in time (i.e. descriptive, not predictive).

In section 2.2.2 of the appendix different methods are presented totreat redundant metabolic models. For the calculation of the fluxes, theoptimization problem (equation 2.9 in the appendix) was solved and thusa weighted least squares solution was obtained. This allowed theextracellular fluxes obtained by solving the optimization problem todiffer from the fluxes calculated directly from the experimental data.

Equation (2.10) of the appendix was used to calculate the extracellular,balanced fluxes, the intracellular ones were determined by equation(2.5) of the appendix, also

{dot over (v)} _(m)=(I−C _(v) _(m) R′ ^(T)(R′C _(v) _(m) R′ ^(T))⁻¹ R′)v _(m)

and

{circumflex over (v)} _(u) =−A _(u) ^(#) A _(m) {circumflex over (v)}_(m).

For the formulation of the covariance matrix C _(v) _(m) , the resultsof the model validation are preferably included, which showed whichstandard deviations led to a largely consistent model and which measuredvalues should possibly be classified as unreliable.

Afterwards the calculated fluxes were visualized: On the one hand, theirtime courses were plotted separately according to metabolites, on theother hand, the entire flux distribution at selected points in time wasvisualized.

FIG. 11 shows snapshots of intracellular and extracellular fluxes atdifferent points in time during the cultivation of a cell culture of CHOcells.

In addition to the course of individual material fluxes over time,snapshots of the entire intra- and extracellular flux distribution inthe individual time intervals may be considered. FIG. 11 shows anexample of four metabolic states of cells from the fourth fed-batchfermentation. They originate from the second, eleventh, 14th and 21sttime intervals. The four pictures show flux distributions during afed-batch fermentation. The gluc39ose flow in the second time intervalserves as a reference. The product flow was multiplied by 10,000.

In the second time interval, i.e. in an early fermentation phase (phaseI in the classification, shown in FIG. 11A, glucose is still in excess,is rapidly transported into the cells and enters the citrate cycle.Glutamine is also taken up, conversion to glutamate takes place andfurther together with pyruvate via flux v_8 into metabolites of thecitrate cycle and into alanine. Lactate and ammonia are released intothe medium in larger quantities.

In the eleventh time interval (phase III), shown in FIG. 11B, thelactate shift has already taken place, i.e. lactate is absorbed from themedium, metabolized to pyruvate and then enters the citrate cycle.Glucose consumption is reduced, glutamine is no longer absorbed.Ammonium is no longer released. Biomass production has decreased, butproduct formation has increased.

Compared to the start of fermentation, the citrate cycle runs withalmost unchanged intensity, but the anaplerotic reaction of malate topyruvate is reduced.

In the 14th time interval (also phase III), shown in FIG. 11C, thelactate concentration in the medium is very low, so that intake is alsorestricted. Therefore alanine is absorbed and converted into pyruvate.Ammonium is again increasingly released into the medium. The productformation has increased even further.

Towards the end of the fermentation (phase IV), shown in FIG. 11D,glucose is added in a bolus-like manner. Therefore, the uptake isincreased again and the flux through the citrate cycle is particularlystrong. However, all other reactions are extremely reduced.

FIG. 12 shows the strongly correlating course of intracellular fluxes,which were calculated for the current point in time using descriptiveMFA and predicted for a point in time in the future using a combinationof the MLP and MFA. A “descriptive MFA for the current point in time”calculates the extracellular fluxes of the metabolic model from themeasured fluxes of the extracellular metabolites over the just elapsedtime interval and uses these extracellular fluxes as input for the MFAto calculate the intracellular fluxes for the current point in time. Incontrast, the “predictive” determination of the intracellular fluxes isbased on predicting extracellular fluxes for the future point in timeusing the trained predictions of the MLP and using the extracellularfluxes thus predicted as input for the NSA to predict the intracellularfluxes at the future point in time.

FIG. 13 shows some of the extracellular metabolites whose concentrationsare used as input values to predict extracellular fluxes.

For each extracellular metabolite flux (given in the top row of thetable) that is to be predicted once by the trained MLP and that ispassed as output parameter values during training, the column belowlists the input metabolites in descending relevance for estimating theflux of the “output metabolites”. Bio” here means the measured biomass,preferably specified in terms of cell density, and “TZD” means the valuecorrected for LDH concentration. The PMI calculation performed todetermine this relevance is preferably based on the values from alltraining fermentations performed.

The table also contains the results of the cross-validation, where thenumber of input variables, the number of hidden variables H and thenumber of iterations were determined. The input metabolites 1504, whoseconcentrations enter the neural network as input parameter values, arehighlighted in yellow.

The values from several (e.g. 3) fed-batch fermentations as well asthose from one batch approach were used as training data set. Data fromother fermenters with the same cell type formed the test data set. Inaddition, cross-validations with other training/test data setpartitioning may be performed.

However, the listing shown in the table does not necessarily correspondto an order that is intuitive from a biological point of view: forexample, glucose concentration plays a subordinate role here, althoughit would be obvious that the most important substrate of the CHO cellshas a major influence on some substance conclusions. However, thePMI-based arrangement described below has only limited biologicalsignificance, which may be attributed to the following facts: thecourses of metabolite concentrations are correlated to a certain extentvia metabolism. Therefore, it is possible that after the selection ofone metabolite, many others may lose relevance for the estimation, sincemuch of the information contained in them is already described by thefirst metabolite. However, the selected metabolite may not be the onethat, from a mechanistic point of view, actually has an influence on thesubstance flow, but is only strongly correlated with it. In fact, it mayhave been observed that a sample of white change in the data selectionused to calculate the PMI values resulted in different arrangements insome cases. In some cases, for example, the position of glutamate andglutamine, which are closely linked via the metabolism, was reversed.However, a selection based on biological intuition, as it is done todayin some publications, sometimes resulted in much worse predictions. Thismay be due to the fact that the biological relationships are often notalways known and that redundant information is selected. In addition,the literature study to select suitable input parameter values based onpresumed biological relevance takes a lot of time.

At first, for each metabolite flow the potential inputs were sortedaccording to their relevance using Partial Mutual Information (PMI), seeequation 2.19 in the appendix. The PMI was calculated using thediscretization according to Equation 2.22 in the Appendix). The coredensity estimator used is based on city block function, theNadaraya-Watson estimator (see Equation 2.27 in the Appendix) was usedto calculate the residuals. A training data set was used for this. As inthe present case, this may comprise the measurement data of 8 cellculture projects in 8 different bioreactors.

The order of the inputs for estimating the j-th extracellular metaboliteflux was performed according to the following algorithm:

-   -   1. summarize in the set X all potential input variables here all        current concentrations of extracellular metabolites. The set U        contains all already selected input variables (it is empty at        the beginning). Y is the output variable, i.e. the j-th        extracellular metabolite flux between the current and the future        sampling time.    -   2. calculate an approximation for the partial transinformation        between each potential input variable in X and Y considering the        elements in U based on the given, standardized data set. For        example, through standardization, all input variables and the        output variable had a mean value of 0 and a standard deviation        of 0.5, thus eliminating distorting effects on the relevance of        the variables due to different orders of magnitude. Preferably,        this standardization is performed before training the neural        network with respect to the measured values of the training data        set as well as when entering current measured values in the test        procedure using a trained MLP with respect to the currently        obtained measured values.    -   3. note the variable in X, which has the highest PMI value. Add        it to U and remove it from X.    -   4. Repeat steps 2 and 3 until X is empty.

According to embodiments of the invention, the PMI for the parameter X,e.g. a determined extracellular metabolite, is calculated with respectto the parameter Y, e.g. another extracellular metabolite, as follows

$\begin{matrix}{{I^{\prime}\left( {X,Y} \right)}:={\int{\int{{g_{x^{\prime},y^{\prime}}\left( {x^{\prime},y^{\prime}} \right)}{\ln \left( \frac{g_{x^{\prime},y^{\prime}}\left( {x^{\prime},y^{\prime}} \right)}{{g_{x^{\prime}}\left( x^{\prime} \right)}{g_{y^{\prime}}\left( y^{\prime} \right)}} \right)}{dx}^{\prime}{dy}^{\prime}}}}} & (2.19)\end{matrix}$

given with the residues

x′:=x−

[x|U]  (2.20)

and

y′:=y−

[y|U].  (2.21)

Where g is the density function of the marginal or common distributions.The residuals contain only the information of X and Y, which are not yetcontained in U. The larger the value for I′, the stronger thedependence.

An approximate, discrete version of expression 2.19) is as follows:

$\begin{matrix}{{{I^{\prime}\left( {X,Y} \right)} = {\frac{1}{N}{\sum\limits_{n = 1}^{N}{\ln \left( \frac{g_{x^{\prime},y^{\prime}}\left( {x^{{(n)}^{\prime}},y^{{(n)}^{\prime}}} \right)}{{g_{x^{\prime}}\left( x^{{(n)}^{\prime}} \right)}{g_{y^{\prime}}\left( y^{{(n)}^{\prime}} \right)}} \right)}}}},} & (2.22)\end{matrix}$

(x^((n)), y^((n))), n=1, . . . , N, pairs of samples of X and Y and(x^((n)′), y^((n)′)) are the associated, g_(x′), y′-distributed pairs ofresiduals. The usually unknown density functions may in turn beapproximated by core density estimators, which also use information fromthe N samples. These estimators provide—in simple terms—a continuousdensity function which is similar to the histogram of the samples. Itresults from a weighted superposition of N core functions. These in turnare density functions that are bell-shaped and symmetrical about one ofthe sample values each.

Among other things, the Gauss core and the city block function have beenused in publications to date to calculate the PMI.

In general terms, the estimation of the density of a q-dimensionalrandom vector X with the samples x⁽¹⁾, . . . , x^((N)) using the corefunction

with bandwidth μ is

g ^ x  ( x ) = 1 N  ∑ n = 1 N  μ  ( x - x ( n ) ) = 1 N  ∑ n = 1 N 1 μ q   ( x - x ( n ) μ ) .

With the city block function as the core function, this results in:

$\begin{matrix}{{{\hat{g}}_{x}(x)} = {{\frac{1}{N}{\sum\limits_{n = 1}^{N}{\frac{1}{\mu^{q}}{\prod\limits_{j = 1}^{q}\left( {\frac{1}{2}e^{- \frac{{x_{j} - x_{j}^{(n)}}}{\mu}}} \right)}}}} = {\frac{1}{{N\left( {2\; \mu} \right)}^{q}}{\sum\limits_{n = 1}^{N}{e^{{- \frac{1}{P}}{\sum_{j = 1}^{q}{{x_{j} - x_{j}^{(n)}}}}}.}}}}} & (2.23)\end{matrix}$

The common density distribution of two random vectors X and Y can beformulated using an estimator with product core

_(μ) _(x) _(,μ) _(y) (x,y)=

_(μ) _(x) (x)·

_(μ) _(y) (y):

g ^ x , y  ( x , y ) = 1 N  ∑ n = 1 N  μ x  ( x - x ( n ) ) · μ y ( y - y ( n ) ) .

The choice of the range has a significant impact on the quality of theestimate. The larger the bandwidth, the smoother, but also less detailedthe density approximation. In several studies and in the present casethe choice

$\mu = \left( \frac{4}{N\left( {q + 2} \right)} \right)^{\frac{1}{q + 4}}$

proven a success.

For the calculation of redundancies according to equations 2.20 and2.21, the conditional expectation value

[X|U=u] for two random vectors X and U must generally be estimated. TheNadaraya-Watson estimator may be used for this purpose. This is based onthe previously applied principles and can be derived as follows:

  [ X | U = u ] =  ∫ x · g x , u  ( x , u ) g u  ( u )  dx ≈  ∫ x ∑ n = 1 N  ( μ x  ( x - x ( n ) )  μ u  ( u - u ( n ) ) )  dx ∑ n= 1 N  μ u  ( u - u ( n ) ) =  ∑ n = 1 N  μ u  ( u - u ( n ) )  ∫x  μ x  ( x - x ( n ) )  dx ∑ n = 1 N  μ u  ( u - u ( n ) ) =  ∑ n= 1 N  x ( n )  μ u  ( u - u ( n ) ) ∑ n = 1 N  μ u  ( u - u ( n )) .

In the approximate step the densities were approximated by means of coredensity estimators. The last equal sign results from the fact that

_(μ) _(x) (⋅−x^((n))) is.

According to embodiments of the invention, the selection is carried outaccording to the principle of a wrapper: For each output variable 70nets with the 1 to 7 (according to PMI) most relevant inputs and with 1to 10 hidden neurons were trained in initially 1000 iterations. For thispurpose, a training data set was created from a part of the entiretraining data set, which was formed from the 8 monitored trainingfermentations. The remaining data served as test data. For each trainingsession, the value of the test error E was recorded over the number ofiterations and its minimum value and the corresponding number ofiterations were determined. Subsequently, the 70 nets were comparedusing the total minimum test error. This resulted in the determinationof the combination of input variables to estimate the respectivemetabolite flux, as well as the corresponding number of iterations andthe number of hidden neurons.

The list of selected input parameter values, here also called “inputs”or “input variables”, for different output parameter values (“outputs”,“output variables”) is indicated accordingly in FIG. 13.

According to preferred embodiments, the selection of those extracellularmetabolites whose concentration is to be used as input parameter valuesfor training or feeding the trained MLP (“input metabolites”) is madeaccording to purely statistical criteria, individually for each outputmetabolite, i.e. individually for each extracellular flux to bepredicted.

1. Ranking the Input Metabolites According to their “Relevance”:

First, all measurable available input metabolites or at least allmetabolites that occur in the metabolic model as extracellularmetabolites are transferred into a “first set” and sorted according totheir relevance: By using the PMI criterion, it is determined whichmetabolite has the greatest significance/predictive power for the output(rate of the extracellular metabolite whose extracellular flux is to bepredicted—“output metabolite”). This metabolite is transferred to a“second set”, which then provides the actual input parameter values. The“relevance” or predictive power determined in this first sorting stepdoes not yet depend on the metabolites in the second set.

2. Determination of the True Input Metabolites:

Preferably, not all input metabolites are used as input for MLP trainingor MLP application (this leads to very poor predictive power due tooverfitting). It is therefore determined how many of the most relevantinput metabolites should be used. This is done by determining thepredictive power of the “x” most relevant input metabolites from a testdata set, varying x, and then selecting the number x with the bestpredictive power.

After the first sorting step and the transfer of the most relevantmetabolite from the first to the “second set” of actual inputmetabolites (whose concentration is provided by the input parameters ofthe MLP), the input metabolite within the remaining metabolites in thefirst set is repeatedly identified that has the greatest predictivepower with respect to the flux of a determined output metabolite, takinginto account the content of the second set. If the concentration profileof the metabolite with the highest predictive relevance within theremaining members of the first set correlates strongly with a metabolitealready contained in the second set, this metabolite is usually nottransferred to the second set, since although its predictive power maybe high, its concentration profile does not make a significantcontribution over that of a metabolite already contained in the secondset. Rather, its uptake would only increase the amount of redundantinformation in the second set. Therefore, if the metabolite in the firstset may not be included in the second set for these reasons, themetabolite with the next highest relevance score of the first set, whichdoes not lead to an excessive increase in the redundancy of theinformation content of the concentration profiles of the metabolites ofthe second set, is transferred from the first to the second set.

Thus, a metabolite may be very meaningful for the rate of output of acertain output metabolite without being transferred to the second set. Atransfer may be omitted in particular if the concentration profile ofthis metabolite correlates very strongly with that of a metabolite thatis already contained in the second set, so that it is sufficient to useonly one of the two or its concentrations as an input parameter valuewhen training the MLP and later also when using the trained MLP. Thismeans that if one of the two is selected as “relevant”, the other losesits significance. This is recognized by the PMI criterion). Continue inthis way until all input metabolites that are relevant in terms of theirpredictive power and sufficiently independent of each other have beenincluded in the second set.

For example, in the case of the Glu flux (4th column in the table shownin FIG. 13) it has been found that information about the current biomassand the current glycine concentration may provide better predictionsthan if only biomass is used as input (too little information), or ifadditional information about glutamine, aspartate etc. is fed into theneural network (too much information, overfitting).

FIG. 14 shows a histogram of the obtained “Root Mean Square Error”(RMSE) for intracellular fluxes obtained for several fed-batchfermentation runs: 12 fed-batch reactors were prepared, each containingone CHO cell clone different from the cell clones of the 11 otherreactors. The 12 cell clones were genetically modified so that all 12clones produced the same product, namely a bispecific antibody. Althoughthe same DNA sequence was used during transfection to produce theclones, the clones exhibit metabolic differences due to differentinsertion loci and/or different copy numbers of the integrated DNAsequences. 10 fed-batches were used for training the MLP (here: a neuralnetwork model), two fed-batches for testing the model. These twofed-batches were used for the RMSE calculations.

The RMSE for the intracellular fluxes are calculated from a differencebetween the intracellular fluxes predicted by a combination of MLP andMFA and intracellular fluxes calculated from measured extracellularfluxes. RMSE is never negative, a value of 0 (almost never reached inpractice) would indicate a perfect fit of predicted and measured data.In general, a lower RMSE is better than a higher one. RMSE is the squareroot of the average of the squared errors. The effect of each error onRMSE is proportional to the magnitude of the squared error, so largererrors have a disproportionate effect on RMSE.

The same type of medium and the same culture media were used for the 12fed-batch reactors to produce the bispecific antibody. However, themedium and culture media differed from the medium and culture media usedfor the bioreactors or cell cultures, whose metabolic footprint is shownin FIG. 5.9 of the Appendix.

The MLP, here a neural network (NN), was recalibrated to the new dataset, i.e. the NN, which had already been trained once on data obtainedfrom the bioreactors shown in FIG. 5.9 of the Appendix, was “retrained”or newly trained on data from the 12 fed-batch reactors for—productionof the bispecific antibody. The models were generated in Python usingthe machine library—Scikit learn. The hyperparameters of various NNmodels were optimized using a grid search function. Models andhyperparameters that best matched the data were stored and used in theform of a “re-trained” NN for future predictions of extracellular aswell as intracellular fluxes of the cell cultures in the 12 bioreactors.

FIG. 15 shows a histogram of the obtained RMSE for extracellular fluxesobtained for the 12 fed-batch fermentation runs mentioned in FIG. 14.All RMSE values are normalized to the error obtained for the metaboliteglucose. The RMSE values for extracellular fluxes are calculated fromthe difference between measured external fluxes and extracellular fluxespredicted by a combination of the MLP, in particular a neural network,and MFA.

It may be observed that the RMSEs of both intracellular andextracellular fluxes obtained for the 12 fed-batch reactors for theproduction of the bispecific antibody were in the same range as RMSEsobtained for other cell cultures or other cell clones (see MasterThesis—“Appendix”—page 67, FIG. 5.9, the protein product of this cellculture is an antibody fusion protein. The 12 cell cultures are CHO cellcultures. The RMSE of the external and internal fluxes (measured againstthe predicted ones) were between 10-35%.

FIG. 16 shows a comparison of the predicted extracellular metabolitefluxes (black line) with two extracellular metabolite fluxes (two greylines) measured for two identical cell clones in different bioreactors(fed-batch and split batch bioreactor) for different metabolites (aminoacids). The cell clones in both bioreactors are recombinant CHO cells(monoclonal), which synthesize a specific antibody fusion protein. Thecurves show that the method according to embodiments of the invention iscapable of very accurately predicting the fluxes of extracellularmetabolites for both fed batch (fb) and split batch (b). In detail, FIG.16 shows a comparison of the extracellular measured fluxes with thepredicted extracellular fluxes in different combinations of trainingdata/test data set. The expression “fb/fb+b” means fed-batch data set astraining data and fed-batch+batch as test data set. One run from thetest data set is then shown in each case.

FIG. 17 shows several plots with two curves each, all obtained for a fedbatch bioreactor with a cell clone for the production of the antibodyfusion protein. The curves consisting of dotted lines (“descriptiveMFA”) were obtained by using measured concentrations of extracellularmetabolites as input of a metabolic flux analysis (MFA) to calculatedifferent extracellular fluxes, each corresponding to one of the 11plots. The solid line curves (“NN-MFA”) were obtained by using measuredconcentrations of extracellular metabolites at a determined point intime t0 as input of an MLP (e.g. NN) to predict extracellular fluxes ata future point in time t1, and using these future predictedextracellular fluxes as input for the metabolic flux analysis (MFA).Thus, a comparison of the two curves of each of the 11 plots shows thatthe values predicted by MLP+MFA for a future point in time have a veryhigh agreement with values obtained with a static MFA model usingmeasured now-time concentrations of extracellular metabolites.

FIG. 18 shows several plots with two curves each, all obtained for a fedbatch bioreactor with a first cell clone ZK1 for the production of abispecific antibody. The curves consisting of dotted lines(“NN+Descriptive MFA extracellular”) were obtained by using measuredfluxes of extracellular metabolites at a particular point in time t0 asinput of an MLP (e.g. NN) to predict extracellular fluxes at a futurepoint in time t1 and using these future predicted extracellular fluxesas input for the metabolic flux analysis (MFA). The crossed line curves(“Descriptive extracellular MFA”) were obtained by using currentlymeasured extracellular fluxes of extracellular metabolites as input formetabolic flux analysis (MFA) to calculate different extracellularfluxes, each corresponding to one of the plots shown in FIG. 18. Theextracellular fluxes were normalized for each plot and for each day interms of glucose concentration. Thus, instead of measured extracellularmetabolite concentrations, current extracellular fluxes calculated basedon current and past metabolite concentrations at a point in time in thepast were used as MLP input. Thus, a metabolite concentration in abroader sense was used as input. It was observed that if alternativelythe measured metabolite concentrations were used as input, theprediction results of the MLP were ultimately essentially identical.Thus, measured concentrations in the narrower sense as well asmetabolite concentrations in the broader sense may equally serve asinput for the MLP.

Thus, a comparison of the two curves of each of the plots in FIG. 18shows that the values predicted by MLP+MFA for a future point in timeare in good agreement with values obtained with a static MFA model usingmeasured now-time concentrations of extracellular metabolites. However,deviations in detail are possible.

The plots in FIG. 18 compare a) the progress of the measured(extracellular) fluxes with their MLP predicted counterparts. All fluxesare expressed as normalized values—normalized to the measured flux.Therefore the glucose flux is always at 1 (not shown as a plot); and b)the progress of the internal fluxes from measured external fluxescalculated by MFA with the internal fluxes from external fluxescalculated by MFA from external fluxes calculated by NN. Again, allfluxes are normalized against the glucose flux.

For individual metabolites, the measured fluxes occasionally deviatedfrom the predicted ones. On the one hand, however, it should be notedhere that the scaling corresponds to a very high “resolution” due to theglucose normalization and the deviations were rather small whenconsidering the total amount of metabolite. Furthermore, a certaintendency of the data to overfitting was observed, which can usually becorrected by increasing the size of the data set.

The plots of FIG. 18 generated for 11 different extracellular fluxesrepresent a “metabolic fingerprint” of the cell clone ZK1. In thefollowing, it will be shown that this fingerprint may differsignificantly from that of other cell clones that produce the sameproduct (bispecific antibodies, but which contain the sequence codingfor the protein in a different number of copies or at a differentlocation in the genome.

FIG. 19 shows several plots with two curves each, all obtained foranother fed batch bioreactor BR2 with a second cell clone ZK2 for theproduction of the bispecific antibody. The curves were determined asdescribed for FIG. 18, but based on data from the second cell clone ZK2.The plots shown in FIGS. 18 and 19 are based on data from twobioreactors BR1, BR2, operated with the same type of medium and culturemedium and which differ essentially only in that the two bioreactorshold different cell clones ZK1 and ZK2, which both synthesise the sameproduct (bispecific antibody) but have genetic differences which mayalso affect the metabolism of the clones: For example, the clones mayhave been produced by a method which does not provide complete controlover the position of integration of a new DNA sequence segment into thegenome of the cells and/or the number of integrated sequence segments.Thus, although the starting cells used are genetically identical, in thecourse of the integration of new genes or DNA sequences (e.g. in thecase of random integration in the course of a transfection), the genesequences for the light chain (LC) and the heavy chain (HC) of thebispecific antibody may be inserted in different numbers of copies anddifferent genome loci. This in turn may cause metabolic differences andalso differences in the productivity and vitality of a specific cellclone, which according to the embodiments of the invention are readilydiscernible and differentiable by metabolic predictions of the MLP andMFA combination. The second bioreactor BR2 was identical in constructionto bioreactor BR1, which was used for the cell culture of clone ZK1.

FIGS. 18 and 19 each show for a specific cell clone ZK1, ZK2 that thecombination of a neural network NN and a metabolic flux analysis MFAleads to different and ultimately more accurate predictions than aprediction of extracellular fluxes based purely on MFA and actual time.In addition, this combination has the advantage that the quality of theprediction can be repeatedly verified by the measured fluxes. A directprediction of the intracellular fluxes would only allow a verificationwith 13C-labelled metabolites, which is neither practicable noreconomical, especially for high throughput or even productionfermentations. Furthermore, a calculated amount of predictedextracellular fluxes for a specific cell clone ZK1, ZK2, as e.g. shownin FIG. 18 for ZK1 and in FIG. 19 for ZK2, represents a “metabolicfingerprint” of a specific cell clone. This may be analysed to identifyspecific cell clones which appear to be particularly beneficial withrespect to a determined parameter such as growth rate, rate of ammoniadegradation, rate of serine production, etc. Metabolic “fingerprints”may also be used to select or characterise two or more different cellclones with regard to their metabolic similarity.

FIG. 20 shows several plots with two curves each, which were allobtained for the BR1 bioreactor with the cell clone ZK1 and which eachrepresent calculated intracellular fluxes. FIG. 20 corresponds to FIG.18 with the difference that the calculated intracellular instead ofextracellular fluxes are shown.

FIG. 21 shows several plots with two curves each, which were allobtained for the BR2 bioreactor with the cell clone ZK2 and which eachrepresent calculated intracellular fluxes. FIG. 21 thus corresponds toFIG. 19 with the difference that the calculated intracellular instead ofthe calculated extracellular fluxes are shown. The entirety of the plotsin FIGS. 19 and 21 thus represents a metabolic fingerprint which may beused for the metabolic characterization of a cell clone. By collectingdata from a large number of these “fingerprints” for a large number ofdifferent clones and fermentation conditions and by recording thevitality and/or productivity of the individual clones, it is possible,e.g. by means of correlation analyses, to identify advantageousmetabolic fingerprints and to use these metabolic fingerprintsidentified as advantageous as a reference value in order to identifyadvantageous clones for future cloning and to select them for afermentation project.

LIST OF REFERENCE NUMERALS

-   -   102-112 Steps    -   200 System    -   202 Processor    -   204 Bioreactor    -   206 Bioreactor    -   208 Bioreactor    -   210 Measured value interface    -   212 Storage medium    -   214 Metabolic model    -   216 Reference values    -   218 Machine learning logic    -   220 Program logic    -   222 Control interface    -   224 User interface    -   250 Device for determining the concentration of metabolites    -   252 Computer system for calculation and prediction    -   254 Metabolic model with fluxes    -   256 Measured extracellular metabolite concentrations    -   402 Metabolic model    -   404 Extracellular metabolites    -   406 Intracellular metabolites    -   408 Extracellular fluxes    -   410 Intracellular fluxes    -   502 Course of measured concentrations extracellular metabolite    -   504 Course of predicted extracellular flux    -   506 Course of intracellular fluxes calculated by MFA    -   802 Plot for Fed-Batch Bioreactor 1    -   804 Plot for Fed-Batch Bioreactor 5    -   806 Plot for Batch Bioreactor    -   808 Plot for Split-Batch Bioreactor    -   1502 Output parameter values    -   1504 Input parameter values

1. A method for predicting the metabolic state of a cell culture ofcells of a specific cell type, comprising: providing a metabolic modelof a cell of the specific cell type, the metabolic model including aplurality of intracellular and extracellular metabolites and a pluralityof intracellular and extracellular fluxes, the metabolic modelcomprising stoichiometric equations specifying at least onestoichiometric relationship between one of the intracellular and one ofthe extracellular metabolites; at each of a plurality of points in timeduring cultivation of the cell culture: receiving a plurality ofmeasurement values measured at said point in time, said measurementvalues comprising concentrations of a plurality of extracellularmetabolites of the metabolic model in the culture medium of the cellculture and a measured cell density of the cells in the cell culture;inputting the received measured values as input parameter values into atrained machine learning program logic—MLP; predicting extracellularfluxes of the extracellular metabolites at a future point in time by theMLP using the received measurement values, the future point in timebeing a point in time subsequent to the point in time of receiving themeasurement values, wherein the extracellular fluxes are uptake rates ofthe extracellular metabolites into a cell and/or release rates of theextracellular metabolites from a cell into the medium; performingmetabolic flux analysis to calculate the intracellular fluxes at thefuture point in time using the predicted extracellular fluxes of theextracellular metabolites and the stoichiometric equations of themetabolic model.
 2. The method according to claim 1, further comprisinga generation of MLP by machine learning, wherein said generationcomprises: generating a training data set, wherein said generatingcomprises: at each of a plurality of training points in time during thecultivation of at least one training cell culture of cells of thespecific cell type: receiving a plurality of measurement values measuredat said training point in time, said measurement values comprisingconcentrations of a plurality of extracellular metabolites of themetabolic model in the culture medium of said at least one training cellculture and a measured cell density of the cells in said at least onetraining cell culture; receiving the time indication of the currenttraining point in time; and calculating extracellular fluxes of theextracellular metabolites as a function of the measured values receivedat that point in time and the measured values received at the respectivepreceding point in time, wherein the extracellular fluxes are uptakerates of the extracellular metabolites into the cell and/or releaserates of the extracellular metabolites into the medium; training theMLP, wherein the training comprises: inputting the measured valuesreceived at each of the training points in time as input parametervalues to the MLP, and inputting the extracellular fluxes calculated forthat following point in time at each point in time following thattraining point in time as output parameter values associated with thoseinput parameter values to the MLP; and performing a learning process bythe MLP in such a way that the MLP learns to predict the respectiveassociated output parameter values based on the input parameter values;storing the trained MLP in a volatile or non-volatile storage medium. 3.The method according to claim 2, wherein the training data set isgenerated such that at each of a plurality of training points in timeduring the cultivation of multiple training cell cultures of cells ofthe specific cell type, the measured values and time specifications arereceived and the extracellular fluxes of the extracellular metabolitesare calculated, wherein the cell cultures are cultivated in bioreactorsof different types, wherein the types of bioreactors comprise at leasttwo different bioreactor types from the following bioreactor types: afed-batch bioreactor, a batch bioreactor, a perfusion reactor, achemostat and a split-batch bioreactor.
 4. The method according to claim1, wherein the MLP is a neural network or a system of several neuralnetworks.
 5. The method according to claim 1, wherein the measuredconcentrations of the multiple extracellular metabolites are each: anindication of the volume-related content of the metabolite, inparticular a mass concentration or a substance concentration, or a valuewhich correlates in a linear manner or at least approximately linearlywith the volume-related content, in particular a corrected or normalisedmeasured metabolite concentration or a measured flux of theextracellular metabolite.
 6. The method according to claim 1, whereinthe MLP is a system of several sub-MLPs, wherein the individual sub-MLPscontained in the system have each been trained to predict theextracellular flux of a single extracellular metabolite and areselectively used to predict the extracellular flux of this singleextracellular metabolite at the future point in time.
 7. The methodaccording to claim 1, wherein the MLP uses measured concentrations of aplurality of extracellular metabolites as input parameter values forpredicting the extracellular flux of a single one of the extracellularmetabolites, wherein the plurality of extracellular metabolites used asinput parameter values differ for at least two of the extracellularmetabolites whose extracellular flux is to be determined.
 8. The methodaccording to claim 7, further comprising: measuring the concentrationsof all input candidate metabolites over a plurality of points in time,wherein the input candidate metabolites comprise all extracellularmetabolites that are measurably available in a reference bioreactor witha cell culture of the specific type or comprise all extracellularmetabolites of the metabolic model; for each of the extracellularmetabolites whose extracellular flux is to be predicted, performing aselection procedure to identify the plurality of extracellularmetabolites to be used as input parameter values for predicting theextracellular flux of that single metabolite, the selection procedurecomprising, with respect to that single metabolite, respectively (a)defining a first set of extracellular metabolites, the first setcomprising all candidate input metabolites; (b) calculating a firstrelevance score of each of the extracellular metabolites in the firstset as a function of the measured concentrations of that metabolite, thefirst relevance score indicating the predictive power of theconcentration of the respective extracellular metabolite with respect tothe extracellular flux of that single extracellular metabolite; (c)transferring only that one of the extracellular metabolites having thehighest first relevance score from the first set to a still empty secondset of extracellular metabolites, removing this metabolite from thefirst set; d) calculating a further relevance score of each of theextracellular metabolites in the first set as a function of the measuredconcentrations of that metabolite and the measured concentrations of allextracellular metabolites contained in the second set, the furtherrelevance score indicating the predictive power of the concentration ofthe respective candidate input metabolite with respect to theextracellular flux of that single extracellular metabolite taking intoaccount the metabolites already contained in the second set; (e)transferring only that one of the extracellular metabolites of the firstset which has the highest further relevance score to the second set,removing this metabolite from the first set, the transfer taking placeonly if, by the inclusion of this metabolite, the second set does notexceed a maximum limit for informative redundancy of the metabolitescontained therein with respect to the prediction of the extracellularflux of this single extracellular metabolite; (f) repeating steps d) ande) until no more metabolites can be transferred from the first to thesecond set without the second set exceeding the maximum informativeredundancy limit; and (g) using selectively only the metabolitestransferred to the second set as input parameter values to predict theextracellular flux of that single extracellular metabolite.
 9. Themethod according to claim 8, wherein the first relevance score iscalculated as a partial mutual information score—PMI score—between ametabolite of the first set and the single metabolite whoseextracellular flow is to be predicted; wherein the second relevancescore is calculated as a PMI score—between a metabolite of the first setand the individual metabolite whose extracellular flow is to bepredicted, taking into account all the metabolites already contained inthe second set.
 10. The method according to claim 1, wherein the MLPuses measured concentrations of a plurality of extracellular metabolitesas input parameter values for predicting the extracellular flux of eachof the individual extracellular metabolites, wherein the plurality ofextracellular metabolites comprises at least one, preferably at leasttwo amino acids.
 11. The method according to claim 1, wherein theplurality of points in time are separated by time intervals of 10minutes to 48 hours, preferably 1-24 hours.
 12. The method according toclaim 1, wherein the cell type is a genetically modified cell type whichis maintained and/or grown in a bioreactor for the purpose of obtaininga biomolecule.
 13. The method according to claim 1, wherein at each ofthe future points in time a calculation of several or all of theintracellular fluxes of the metabolic model is performed.
 14. The methodaccording to claim 1, further comprising: identifying one or moreintracellular metabolites of the metabolic model whose calculatedintracellular fluxes deviate from a respective reference value orreference value range by more than the threshold value; automaticallyidentifying that intracellular flux which acts as a limiting factor forcell growth or production of a desired biomolecule.
 15. A method formonitoring and/or controlling a bioreactor which includes the cellculture of cells of a specific cell type, comprising: calculating theintracellular fluxes of the cells at a future point in time according tothe method according to claim 1; comparing the predicted intracellularfluxes with reference values or reference value ranges for acceptableintracellular fluxes of the respective one or more intracellularmetabolites; issuing a warning if a deviation of the calculatedintracellular flux of at least one of the intracellular metabolites fromits respective reference value or reference value range exceeds a limitvalue; and/or sending a control command to the bioreactor toautomatically initiate steps which change the state of the bioreactor orthe medium contained therein in such a way as to reduce the deviation,wherein the automatic steps include in particular a change in thequantity and/or a change in the composition of a culture medium.
 16. Themethod according to claim 15, further comprising: identifying thereaction within the metabolic model of the cells which acts as alimiting factor for cell growth or production of a desired biomolecule;and automatically adding of selectively those substances which modifythe intracellular flux acting as a limiting factor in such a way as topromote cell growth or production of the biomolecule or the quality ofthe biomolecule, or issuing a request for such addition via a userinterface.
 17. A method for identifying a metabolically favorable cloneof cells of a specific cell type, comprising: culturing different cellcultures in multiple bioreactors, wherein the different cell culturesinclude different clones of cells of the specific cell type; calculatingthe intracellular flux of one or more intracellular metabolites atmultiple points in time separately for each of the cell clones accordingto claim 1; identifying that one of the cell clones whose calculatedintracellular flux of the one or more intracellular metabolites ismetabolically most favorable.
 18. The method according to claim 1,further comprising: calculating the current extracellular flux of one ormore of the extracellular metabolites from the concentrations of theextracellular metabolites measured at the current point in time and atthe previous point in time; performing a further metabolic flux analysisto calculate the current intracellular fluxes at the current point intime using the calculated current extracellular fluxes of theextracellular metabolites and the stoichiometric equations of themetabolic model; and using the calculated current intracellular fluxesas a characterization of a current metabolic state of the cells of thecell culture.
 19. The method according to claim 1, wherein aconcentration of lactate dehydrogenase—LDH—measured in the medium of thecell culture is further received at each of the points in time duringthe cultivation of the cell culture; wherein the prediction of theextracellular fluxes of the extracellular metabolites at each of thefuture points in time by the MLP is made using a corrected rather thanthe measured cell density, wherein the calculation of the corrected celldensity comprises for each of the points in time: calculating thedensity of lysed cells in the medium of the cell culture as a functionof the measured LDH concentration, said function being an empiricallydetermined, heuristic and linear function representing the relationshipof the LDH concentration in the medium to the number of lysed cells ofthat specific cell type; calculating the corrected cell density as thesum of the measured cell density in the medium and the calculateddensity of the lysed cells.
 20. A system for predicting the metabolicstate of a cell culture of cells of a specific cell type, comprising:one or more processors; a first interface for receiving measurementsfrom a bioreactor containing the cell culture; a volatile ornon-volatile storage medium comprising: a metabolic model of a cell ofthe specific cell type, the metabolic model including a plurality ofintracellular metabolites and extracellular metabolites and a pluralityof intracellular and extracellular fluxes, the metabolic model includingstoichiometric equations specifying at least one stoichiometricrelationship between one of the intracellular and one of theextracellular metabolites; a trained machine learning program logic—MLP;a program logic adapted to perform a method at each of a plurality ofpoints in time during the cultivation of the cell culture, the methodcomprising: receiving a plurality of measurement values measured at saidpoint in time via the first interface, the measurement values comprisingconcentrations of a plurality of extracellular metabolites of themetabolic model in the culture medium of the cell culture and a measuredcell density of the cells in the cell culture; inputting the receivedmeasured values as input parameter values to the MLP; predictingextracellular fluxes of the extracellular metabolites at a future pointin time by the MLP using the received measurement values, the futurepoint in time being a point in time subsequent to the time of receipt ofthe measurement values, wherein the extracellular fluxes are uptakerates of the extracellular metabolites into a cell and/or release ratesof the extracellular metabolites from a cell into the medium; performingmetabolic flux analysis to calculate the intracellular fluxes at thefuture point in time using the predicted extracellular fluxes and thestoichiometric equations of the metabolic model.